ee 


Se eae 
pai 


THE UNIVERSITY 
OF ILLINOIS 
LIBRARY 
5 30 
An & 


Ge 


Return this book on or before the 
Latest Date stamped below. A 
charge is made on all overdue 


books. 
U. of I. Library 


| DEC 22°36 
JUL 12 j94 
AR =F 1942 


ae ee 


. a > tee PRR Vy +), Jee? i okt 
Bi bale aly Ree, SU hs a 
7 \) te oe ; : : hy eae t Ae . 
Wy oa cies iT cata 
v3 4 4 a i Mg it 7 rally ¥ 
ha * ey \' tr fad 
ee 4 Oley 
a) Yoh ‘ y 
: 4 
: ? re 
Ws A 
* 
- 
’ 
“ ' 
w ,' . 
\ 
a 
} a E F . 
+ 
‘ ar) " ’ 
J 
Mi | 
it NW ; y 


leat wif) ¢ 
4 ; “ rT 4 
Ol iy , 
Af 
r 


{ie fall Reh) 
" me F aoe ‘ ai vei ‘iy 4a , a) eee hic Bi 
Ao Sa aay Ta 1 AR aN AE ae taal 


eu s 


Renan | : 


ak OK. 


ELEMENTARY 


TEXT-BOOK OF PHYSICS. 


BY 


PROFESSOR WILLIAM A. ANTHONY, 


OF CORNELL UNIVERSITY, 
AND 


me ObESSOR, CYRUS VR BRACKERT, 


OF THE COLLEGE OF NEW JERSEY. 


Peer LOT TION, REVISED AND ENLARGED, 


NEW YORK: 
JOHN WILEY & SONS, 


58 EAST TENTH STREET. 


1890. 


ie «hy, f 


yA N ( , iyi 
a ‘ ; ‘ i ce ee chs (Ne 
‘ ame ic NON 
nA . uF) . wey, 5 oh}, ue 1" 
‘ - 7 Np iy 
i ‘ <a \ oN Nt . a bats 
: ‘ ‘ YW v Fay,“ A 4 
: hh \ ut ' 
et tas re 
t . | 
‘ 
. 
. f “ 
‘ y 
4’ ‘ s 
=e 
4 
j ge ity 
j . 
| 
& 
Copyright, 1887, 
d a 
z J ee: 
By Joun Witey & Sons. — 
a 
~ 4 
.~ 1 


; c nase 
. er Nic on atid 
€ ty 


own é 


“a, ** ” 5 
A ; ie P| 
t> ¥ ei 
‘ rete ys , 


_ - Drummond & Nev, Wo Rye ae 
_ Electrotypers, | | lela 
_ * 3to 7 Hague Street, MYL Calf age 

New York. iol . 


— ae a oo a ai —— 


ci il hdl mall 


Sor 
> CF 


PREFACE. 


THE design of the authors in the preparation of this work 
has been to present the fundamental principles of Physics, the 
experimental basis upon which they rest, and, so far as possible, 
the methods by which they have been established.  Illustra- 
tions of these principles by detailed descriptions of special 
methods of experimentation and of devices necessary for their 
applications in the arts have been purposely omitted. The 
authors believe that such illustrations should be left to the lec- 
turer, who, in the performance of his duty, will naturally be 
guided by considerations respecting the wants of his classes 
and the resources of his cabinet. 

Pictorial representations of apparatus, which can seldom be 
employed with advantage unless accompanied with full and 
exact descriptions, have been discarded, and only such simple 
diagrams have been introduced into the text as seem suited 
to aid in the demonstrations. By adhering to this plan 
greater economy of space has been secured than would other- 
wise have been possible, and thus the work has been kept 
within reasonable limits. 

A few demonstrations have been given which are not usually 


476004. 


iv PREFACE. 


found in elementary text-books, except those which are much 
more extended in their scope than the present work. This has 
been done in every case in order that the argument to which 
the demonstration pertains may be complete and that the stu- 
dent may be convinced of its validity. 

In the discussions the method of limits has been recognized 
wherever it is naturally involved; the special methods of the 
calculus, however, have not been employed, since, in most insti- 
tutions in this country, the study of Physics is commenced be- 
fore the student is sufficiently familiar with them. 

The authors desire to acknowledge their obligations to Wm. 
F. Magie, Assistant Professor of Physics in the College of New 
Jersey, who has prepared a large portion of the manuscript and 
has aided in the final revision of all of it, as well as in reading 
the proof-sheets. 3 

W. A. ANTHONY, 
C. F. BRACKET 
September, 1887 


CONTENTS. 


PAGE 
INTRODUCTION, . é ; : 5 ; : : . ‘ : ; I 
MECHANICS. 
CHAPTER I. MECHANICS OF MASSES, : : ; : : : Aner 
II. MAss ATTRACTION, , ; ; ‘ : 5 , in OF 
III. MoLEcuLAR MECHANICS, ; , ‘ : ; : 2/84 
IV. MECHANICS OF FLUIDS, . : : i ; ; ; 520 
HEAT. 
CHAPTER J. MEASUREMENT OF HEAT, . : : ; ; : duetA3 
Il. TRANSFER OF HEAT, . 3 A : : - . 161 
III. Errects or Hear, 4 A ‘ : : é ‘ et T68 
IV. THERMODYNAMICS, : . : 3 : 3 ; Res 
- MAGNETISM AND ELECTRICITY. 
CHAPTER I. MAGNETISM, . . : : ; ; ‘ 7 : Beers: 
II. ELECTRICITY IN EQUILIBRIUM, . ; ‘ ? ‘ ad 
III. THE ELECTRICAL CURRENT, . , ‘ : : ; i eaze 
IV. CHEMICAL RELATIONS QF THE CURRENT, . 3 ; oe 
V. MAGNETIC RELATIONS OF THE CURRENT, . : , ae 207 
VI. THERMO-ELECTRIC RELATIONS OF THE CURRENT, , 7 SiO 
VII. Luminous EFFECTS OF THE CURRENT, : : : F348 
SOUND. 
CHAPTER I. ORIGIN AND TRANSMISSION OF SOUND, ; , t meOS3 
II. SouNps AND Music, _. : : : ‘ : : ‘OS 
III. VIBRATIONS OF SOUNDING BODIEs, : ‘ , ORS wp 
IV. ANALYSIS OF SOUNDS AND SOUND Sa TTOne . A . 380 
V. EFFECTS OF THE COEXISTENCE OF SOUNDS, . : : 1805 


Mee VELOCITY OF SOUND, . : : : : : 3 - 390 


CONTENTS. 


CHAPTER I, 
ae 

III. 

IV. 

V. 

Vil; 


TABLE O71) 
i. 
ETT; 

1 Ve 

V 

VI. 
VII. 
VIII. 
TX 

x 

ake 

Da le 
PUNE E 


LIGHT, 


PROPAGATION OF LIGHT, F y . ‘ - ‘ 2 
REFLECTION AND REFRACTION, . ; . : : ; 
VELOCITY GFSLIGHT. ©. ; , ‘ ; : ; : 
INTERFERENCE AND DIFFRACTION, ; 4 : : 

DISPERSION, . : Z : a atk : : , : 
ABSORPTION AND EMISSION, . : : : s ; 


. DOUBLE REFRACTION AND POLO: : ; : ; 


TABLES. 


UnNITs OF LENGTH, x d : : ? a : : 
ACCELERATION OF GRAVITY, ; , é ; ; ; 
UNITS OF WoRK, . : ; P : , , : ‘ 
DENSITIES OF SUBSTANCES AT 0°, ‘ . A ‘ ; 
UNITS OF PRESSURE FOR ¢ = OSI, 6 : . : ‘ 
ENASTICITY : 5 : : : ; : : 
ABSOLUTE DENSITY OF Ween , , : ; : ; 
DENSITY OF MERCURY, . é é : : : : : 
COEFFICIENTS OF LINEAR EXPANSION, . : ; ‘ : 
SPECIFIC HEATS—WATER AT 0° =1, . - ; 7 
MELTING AND BOILING POINTS, BIC., . : ' 
MAXIMUM PRESSURE OF VAPOR AT VARIOUS Ternr «sien 
CRITICAL TEMPERATURES AND PRESSURES IN ATMOSPHERES, 
AT THEIR CRITICAL TEMPERATURES, OF VARIOUS GASES, 
COEFFICIENTS OF CONDUCTIVITY FOR HEAT IN C, G. S. 
UNITS, : : 
ENERGY PRanaaee BY oes ATION OF I Chane OF Ouran 
SUBSTANCES WITH OXYGEN, : 
ATOMIC WEIGHTS AND COMBINING Nine od 
MOLECULAR WEIGHTS AND DENSITIES OF GASES, 
ELECTROMOTIVE FORCE CF VOLTAIC CELLS, . ‘ : 
ELECTRO-CHEMICAL EQUIVALENTS, : . ‘ : F 
ELECTRICAL RESISTANCE, ‘ : : : : é : 
INDICES OF REFRACTION, ’ ; ‘ : 
WAVE LENGTHS OF Liture-ROweD: S DEteERManaerae 
ROTATION OF PLANE OF POLARIZATION BY A QUARTZ PLATE, 
I MM. THICK, CUT PERPENDICULAR TO AXIS, . | dees 


. VELOCITIES OF LIGHT, AND VALUES OF gz, . : F ‘ 


INTRODUCTION. 


I. Divisions of Natural Science.—Everything which can 
affect our senses we call matter. Any limited portion of mat- 
ter, however great or small, is called a body. All bodies, to- 
gether with their unceasing changes, constitute ature. 

Natural Science makes us acquainted with the properties 
of bodies, and with the changes, or phenomena, which result 
from their mutual actions. It is therefore conveniently divided 
-into two principal sections,—Natural History and Natural 
Philosophy. 

The former describes natural objects, classifies them accord- 
ing to their resemblances, and, by the aid of Natural Philoso. 
phy, points out the laws of their production and development. 
The latter is concerned with the laws which are exhibited in 
the mutual action of bodies on each other. 

These mutual actions are of two kinds: those which leave © 
the essential properties of bodies unaltered, and those which 
effect a complete change of properties, resulting in loss of 
identity. Changes of the first kind are called physical changes; 
those of the second kind are called chemical changes. Nat- 
ural Philosophy has, therefore, two subdivisions,—Physics and 
Chemistry. 

Physics deals with all those phenomena of matter which are 
not directly related to chemical changes. Astronomy is thus a 
branch of Physics, yet it is usually excluded from works like 
the present on account of its special character. 


2 ELEMENTARY PHYSICS. [2 


It is not possible, however, to draw sharp lines of demarca- 
tion between the various departments of Natural Science, for 
the successful pursuit of knowledge in any one of them re- 
quires some acquaintance with the others. 

2. Methods.—The ultimate basis of all our knowledge of 
nature is experience,—experience resulting from the action of 
bodies on our senses, and the consequent affections of our 
minds. 

When a natural phenomenon arrests our attention, we call 
the result an observation. Simple observations of natural. phe- 
nomena only in rare instances can lead to such cumplete 
knowledge as will suffice for a full understanding of them. An 
observation is the more complete, the more fully we appre- 
hend the attending circumstances. We are generally not cer. 
tain that all the circumstances which we note are condttions on 
which the phenomenon, in a given case, depends. In such 
cases we modify or suppress one of the circumstances, and ob- 
serve the effect on the phenomenon. If we find a correspond- 
ing modification or failure with respect to the phenomenon, 
we conclude that the circumstance, so modified, is a condition. 
We may proceed in the same way with each of the remaining 
circumstances, leaving all unchanged except the single one 
purposely modified at each trial, and always observing the ef- 
fect of the modification. We thus determine the conditions 
on which the phenomenon depends. In other words, we bring 
expertment to our aid in distinguishing between the real condi- 
tions on which a phenomenon depends, and the merely acci- 
dental circumstances which may attend it. 

But this is not the only use of experiment. By its aid we 
may frequently modify some of the conditions, known to be 
conditions, in such ways that the phenomenon is not arrested, 
but so altered in the rate with which its details pass before us 
that they may be easily observed. Experiment also often 
leads to new phenomena, and to a knowledge of activities be- 


2] INTRODUCTION. 3 


— 


fore unobserved. Indeed, by far the greater part of our knowl- 
edge of natural phenomena has been acquired by means of ex- 
periment. To be of value, experiments must be conducted 
with system, and so as to trace out the whole course of the 
phenomenon. 

Having acquired our facts by observation and experiment, 
we seek to find out how they are related; that is, to discover 
the daws which connect them. The process of reasoning by 
which we discover such laws is called zuduction. As we can 
seldom be sure that we have apprehended all the related facts, 
it is clear that our inductions must generally be incomplete. 
Hence it follows that conclusions reached in this way are at 
best only probable; yet their probability becomes very great 
when we can discover no outstanding fact, and especially so 
when, regarded provisionally as true, they enable us to predict 
phenomena before unknown. 

In conducting our experiments, and in reasoning upon them, 
we are often guided by suppositions suggested by previous 
experience. If the course of our experiment be in accordance 
with our supposition, there is, so far, a presumption in its favor. 
So, too, in reference to our reasonings: if all our facts are seen 
to be consistent with some supposition not unlikely in itself, 
we say it thereby becomes probable. The term hypothests is 
usually employed instead of supposition. 

Concerning the ultimate modes of existence or action, we 
know nothing whatever; hence, alaw of nature cannot be 
demonstrated in the sense that a mathematical truth is demon- 
strated. Yet so great is the constancy of uniform sequence 
with which phenomena occur in accordance with the laws 
which we discover, that we have no doubt respecting their 
validity. 

When we would refer a series of ascertained laws to some 
common agency, we employ the term ¢heory. Thus we find in 
the “wave theory” of light, based on the hypothesis of a uni- 


4 ELEMENTARY PLY SICS, IE: 


versal ether of extreme elasticity, satisfactory explanations of 
the laws of reflection, refraction, diffraction, polarization, etc. 

3. Measurements.—All the phenomena of nature occur in 

matter, and are presented to us in ¢zme and space. 

| Time and space are fundamental conceptions: they do not 
admit of definition. Matter is equally indefinable: its distinc- 
tive characteristic is its persistence in whatever state of rest or 
motion it may happen to have, and the resistance which it of- 
fers to any attempt to change that state. This property is. 
called zmertza. It must be carefully distinguished. from inac- 
tivity. 

Another essential property of matter is zazpenetrabtlity, or 
the property of occupying space to the exclusion of other 
matter. 

We are almost constantly obliged, in physical science, to 
measure the quantities with which we deal. We measure 
a quantity when we compare it with some standard of the 
same kind. A simple number expresses the result of the com- 
parison. 

If we adopt arbitrary units of length, time, and mass (or 
quantity of matter), we can express the measure of all other 
quantities in terms of these so-called fundamental units. A 
unit of any other quantity, thus expressed, is called a derzved 
wee. 

It is convenient, in defining the measure of derived units, to. 
speak of the ratio between, or the product of, two dissimilar 
quantities, such as space and time. This must always be un- 
derstood to mean the ratio between, or the product of, the 
numbers expressing those quantities in the fundamental units. 
The result of taking such a ratio or product of two dissimilar 
quantities is a number expressing a third quantity in terms of 
a derived unit. 

4. Unit of Length.—The wuzt of length usually adopted in 
scientific work is the centimetre. It is the one hundredth part 


4] INTRODUCTION. 5 


of the length of acertain piece of platinum, declared to be a 
standard by legislative act, and preserved in the archives of 
France. This standard, called the metre, was designed to be 
equal in length to one ten-millionth of the earth’s quadrant. 

The operation of comparing a length with the standard 
is often difficult of direct accomplishment. This may arise 
from the minuteness of the object or distance to be measured, 
from the distant point at which the measurement is to end 
being inaccessible, or from the difficulty of accurately dividing 
our standard into very small fractional parts. In all such cases 
we have recourse to indirect methods, by which the difficulties 
are more or less completely obviated. 

The vernier enables us to estimate small fractions of the 
unit of length with great convenience and accuracy. It con- 
sists of an accessory piece, fitted to slide on the principal scale 
of the instrument to which it is applied. A portion of the ac- 
cessory piece, equal to z minus one or # plus one divisions of 
the principal scale, is divided into z divisions. 
In the former case, the divisions are numbered 
in the same sense as those of the principal scale ; 
in the latter, they are numbered in the opposite 
sense. In either case we can measure a quan- 10 
tity accurately to the one wth part of one of the | 3 
primary divisions of the principal scale. Fig. 1 
will make the construction and use of the ver- 
nier plain. 

tneiie, I, let0, 1, 2, 3... 10 be the di- ™ : 
wistansmon tne -vernier; let 0, I; 2, 3.... 10 
be any set of consecutive divisions on the 
principal scale. 

If we suppose the o of the vernier to be in 
coincidence with the limiting point of the mag- Fig. 1. 
nitude to be measured, it is clear that, from the position 
shown in the figure, we have 29.7, expressing that magnitude 


31 


6 ELEMENTARY PHYSICS. [4 


to the nearest tenth; and since the sixth division of the ver- 
nier coincides with a whole division of the principal scale, we 
have 56, of +, or ~$,, of a principal division to be added; 
hence the whole value is 29.76. 

The micrometer screw is also much employed. It consists 
of a carefully cut screw, accurately fitting in a nut. The 
head of the screw carries a graduated circle, which can turn 
past a fixed line. This is frequently the straight edge of a scale 
with divisions equal in magnitude to the pitch of the screw. 
These divisions will then show through how many revolutions 
the screw is turned in any given trial; while the divisions on 
the graduated circle will show the fractional part of a revolu- 
tion, and consequently the frac: 
tional part of the pitch that must 
be added. If the screw be turned 
through z revolutions, as shown by 
the scale, and through an additional 
fraction, as shown by the divided 
circle, it will pass through z times. 
the pitch of the screw, and an ad- 
ditional fraction of the pitch deter- 
mined by the ratio of the number 
of divisions read from o on the di- 
vided circle to the whole number 
into which it is divided. 

The cathetometer is.used for 
measuring differences of level. A 
graduated scale is cut on an up- 
right bar, which can turn about a 
vertical axis. Over this bar slide 
two accurately fitting pieces, one 
of which can be clamped to the bar 
at any point, and serve as the fixed bearing of a micrometer 
screw. The screw runs in a nut in the second piece, which has 


\ 


Pica: 


4] INTRODUCTION. 7 


a vernier attached, and carries a horizontal telescope furnished 
with cross-hairs. The telescope having been made accurately 
horizontal by means of a delicate level, the cross-hairs are 
made to cover one of the two points, the difference of level. be- 
tween which is sought, and the reading upon the scale is taken ; 
the fixed piece is then unclamped, and the telescope raised or 
lowered until the second point is covered by the cross-hairs, 
and the scale reading is again taken. The difference of scale 
reading is the difference of level sought. | 
The dividing engine may be used for dividing scales or for 


Bic)'3: 


comparing lengths. In its usual form it consists essentially 
of a long micrometer screw, carrying a table, which slides, 
with a motion accurately parallel with itself, along fixed 
guides, resting on a firm support. To this table is fixed an 
apparatus for making successive cuts upon the object to be 
graduated. | 

The object to be graduated is fastened to the fixed sup- 
port. The table is carried along through any required dis- 


8 ELEMENTARY PHYSICS. [5 


tance determined by the motion of the screw, and the cuts 
can be thus made at the proper intervals. 

The same instrument, furnished with microscopes and ac- 
cessories, may be employed for comparing lengths with a 
standard. It may then be called a comparator. 

The spherometer is a special form of the micrometer screw. 
As its name implies, it is primarily used for measuring the cur- 
vature of spherical surfaces. 

It consists of a screw with a large head, divided into a 
great number of parts, turning in a nut supported on three 
legs terminating in points, which form the vertices of an equi- 
lateral triangle. The axis of revolution of the screw is per- 
pendicular to the plane of the triangle, and passes through its 
centre. The screw ends in a point which may be brought 
into the same plane with the points 
of the legs. This is done by plac- 
ing the legs on a truly plane sur- 
face, and turning the screw till its 
point is just in contact with the sur- 
face. The sense of touch will en- 
able one to decide with great nicety 
when the screw is turned far enough. 
If, now, we note the reading of the 
divided scale, and also that of the 
divided head, and then raise the 

ls screw, by turning it backward, so 
that the given curved surface may exactly coincide with the 
four points, we can compute the radius of curvature from the 
difference of the two readings and the known length of the 
side of the triangle formed by:the points of the tripod. 

5. Unit of Time.—The wnt of time is the mean time 
second, which is the gg}yy of a mean solar day. We employ 
the clock, regulated by the pendulum or the chronometer 
balance, to indicate seconds. The clock, while sufficiently ac 


7] INTRODUCTION. 9 


curate for ordinary use, must for exact investigations be fre- 
quently corrected by astronomical observations. 

Smaller intervals of time than the second are measured by 
causing some vibrating body, as a tuning-fork, to trace its 
path along some suitable surface, on which also are recorded 
the beginning and end of the interval of time to be measured. 
The number of vibrations traced while the event is occurring 
determines its duration in known parts of a second. 

In estimating the duration of certain phenomena giving rise 
to light, the revolving mirror may be employed. By its use, 
with proper accessories, intervals as small as forty billionths of 
a second have been estimated. 

6. Unit of Mass.—The wzzt of mass usually adopted in 
scientific work is the gram. It is equal to the one thousandth 
part of a certain piece of platinum, called the kzlogram, pre- 
served as a standard in the archives of France. This standard 
was intended to be equal in mass to one cubic decimetre of 
water at its greatest density. 

Masses are compared by means of the dalance, the con- 
struction of which will be discussed hereafter. 

7. Measurement of Angles.—Angles are usually measured 
by reference to a divided circle graduated on the system of 
division upon which the ordinary trigonometrical tables are 
based. A pointer or an arm turns about the centre of the 
circle, and the angle between two of its positions is measured 
in degrees on the arc of the circle. For greater accuracy, the 
readings may be made by the help of avernier. To facilitate 
the measurement of an angle subtended at the centre of the 
circle by two distant points, a telescope with cross-hairs is 
mounted on the movable arm. 

In theoretical discussions the unit of angle often adopted 
is the radzan, that is, the angle subtended by the arc of a 
circle equal to its radius. In terms of this unit, a semi-circum- 
ference equals 7 = 3.141592. The radian, measured in degrees, 
feeo ol? 44.9.” 


ite) LLEMEN PALE (PIT Si Ce, [8 


8. Dimensions of Units.—Any derived unit may be repre- 
sented by the product of certain powers of the symbols repre- 
senting the fundamental units of length, mass, and time. 

Any equation showing what powers of the fundamental 
units enter into the expression for the derived unit is called 
its dimensional equation. In a dimensional equation time is 
represented by T, length by L, and mass by M. ‘To indicate 
the dimensions of any quantity, the symbol representing that 
quantity is enclosed in brackets. 

For example, the unit of area varies as the square of the 
unit of length; hence its dimensional equation is [area] = L’. 
In like manner, the dimensional equation for volume is [vol.] 
eile 

9. Systems of Units——The system of units adopted in 
this book, and generally employed in scientific work, based 
upon the centimetre, gram, and second, as fundamental units, 
is called the centimetre-gram-second system or the C. G. S. 
system. A system based upon the foot, grain, and second was 
formerly much used in England. One based upon the milli- 
metre, milligram, and second is still sometimes used in Ger- 
many. . 


MECHANICS. 


CAR Se. 
MECHANICS OF MASSES. 


10. THE general subject of motion is usually divided, in 
extended treatises, into two topics,— Kinematics and Dy- 
namics. In the first are developed, by purely mathematical 
methods, the laws of motion considered in the abstract, inde- 
pendent of any causes producing it,and of any substance in 
which it inheres; in the second these mathematical relations 
are extended and applied, by the aid of a few inductions drawn 
from universal experience, to the explanation of the motions 
of bodies, and the discussion of the interactions which are the 
occasion of those motions. 

For convenience, the subject of Dynamics is further divided 
into S¢atics, which treats of forces as maintaining bodies in 
equilibrium and at rest, and Kzmet¢zcs, which treats of forces as 
setting bodies in motion. 

In this book it has been found more convenient to make 
no formal distinction between the mathematical relations of 
motion and the application of those relations to the study of 
forces and the motions of bodies. The subject is so extensive 
that only those fundamental principles and results will be pre- 
sented which have direct application in subsequent parts of 
the work. 

11. Mass and Density.—In many cases it is convenient to 
speak of the quantity of matter ina body as a whole. It is 
then called the mass of the body. In case the matter is con- 
tinuously distributed throughout the body, its mass is often 


12 ELEMENTARY LLY SICS: [12 


represented by the help of the quantities of matter in its 
clementary volumes. The denszty of any substance is defined 
as the-limit of the ratio of the quantity of matter in any volume 
within the substance to that volume, when the volume is dimin- 
ished indefinitely. In case the distribution of matter in the 
body is uniform, its density may be measured by the quantity 
of matter in unit volume. 

Since density is measured by a mass divided by a volume, 
its dimensions are WL — 

12. Particle.—A body constituting a part of a material 
system, and of dimensions such that they may be considered 
infinitely small in comparison with the distances separating it 
from all other parts of the system, is called a partzcle. 

13. Motion.—The change in position of a material particle 
is called its motzon. It is recognized by a change in the config- 
uration of the system containing the displaced particle; that 
is, by a change in the relative positions of the particles making 
up the system. Any particle in the system may be taken as 
the fixed point of reference, and the motion of the others may 
be measured from it. Thus, for example, high-water mark on 
the shore may be taken as the fixed point in determining the 
rise and fall of the tides; or, the sun may be assumed to be at 
rest in computing the orbital motions of the planets. We can 
have no assurance that the particle which we assume as 
fixed is not really in motion as a part of some larger system ; 
indeed, in almost every case we know that it is thus in motion. 
As it is impossible to conceive of a point in space recognizable 
as fixed and determined in position, our measurements of 
motion must always be relative. | 

One important limitation of this statement must be made : 
by proper experiments it is possible to determine the absolute 
angular motion of a body rotating about an axis. 

14. Path.—The moving particle must always describe a 
continuous line or path. In all investigations the path may be 


15] MECHANICS OF MASSES. 13 


represented by a diagram or model, or by reference to a set of 
assumed co-ordinates. 

15. Velocity.—The rate of motion of a particle is called 
its velocity. If the particle move in a straight line, and de- 
scribe equal spaces in any arbitrary equal times, its velocity is 
constant. A constant velocity is measured by the ratio of the 
space traversed by the particle to the time occupied in travers- 
ing that space. Ifs, and s represent the distances of the par- 
ticle from a fixed point on its path at the instants 7, and 7, 
then its velocity is represented by 


Ui Pda Re (1): 


If the path of the particle be curved, or if the spaces described 
by the particle in equal times be not equal, its velocity is varza- 
ble. The path of a particle moving with a variable velocity 
may be approximately represented by a succession of very 
small straight lines, which, if the real path be curved, will differ 
in direction, along which the particle moves with constant 
velocities which may differ in amount. The velocity in any 
one of these straight lines is represented by the formula 

S— 5S, 
i i vie a 
each of the spaces s — s, will become indefinitely small, and in 
the limit the imaginary path will coincide with the real path. 


v As the interval of time ¢ — ¢, approaches zero, 


So 


= F will represent the velocity of 


errs : S 
The limit of the expression 7 


the particle along the tangent to the path at the time = ~/,, 

or, as it is called, the velocity in the path. This limit is usually 
ds 

expressed by ay 


The practical unit of velocity is the velocity of a body mov- 
ing uniformly through one centimetre in one second. 
The dimensions of velocity are LT™’. 


i4 ELEMENTARY PHYSICS. — <6 


16. Momentum.—The somentum of a body isa quantity 
which varies with the mass and with the velocity of the body 
jointly, and is measured by their product. Thus, for example, 
a body weighing ten grams, and having a velocity of ten centi- 
metres, has the same momentum as a body weighing one gram, 
and having a velocity of one hundred centimetres. The prac- 
tical unit of momentum is that of a gram of matter moving 
with the unit velocity. The formula is 


my, (2) 


where m represents mass. 

The dimensions of momentum are JZZ7~—’. 

17. Acceleration.— When the velocity of a particle varies, 
its rate of change is called the acceleration of the particle. 
Acceleration is either positive or negative, according as the 
velocity increases or diminishes. If the path of the particle 
be a straight line, and if equal changes in velocity occur in 
equal times, its acceleration is coustant. It is measured by the 
ratio of the change in velocity to the time during which that 
change occurs. If v, and wv represent the velocities of the par- 
ticle at the instants ¢, and ¢, then its acceleration is represented 


by 


Vv — U, 


UA means ; (3) 


If the path of the particle be curved, or if the changes in 
velocity in equal times be not equal, the acceleration is varzadle. 
It can be easily shown, by a method similar to that used in the 
discussion of variable velocity, that the limit of the expression 


will represent the acceleration in the path at the 


time ¢=/7,. This acceleration is due to a change of velocity 
in the path. It is not in all cases the total acceleration of the 


17] MECHANICS OF MASSES. 15 


particle. As will be seen in § 37, a particle moving along a 
curve has an acceleration which is not due to a change of 
velocity in the path. 3 

The practical unit of acceleration is that of a particle, the ve- 
locity of which changes by one unit of velocity in one second. 

The dimensions of acceleration are L 77*. 

The space s — s, traversed by.a particle moving with a con- 
stant acceleration /, during a time ¢—Z,, is determined by 
considering that, since the acceleration is constant, the aver- 


age velocity oe: for the time ¢ — ¢,, multiplied by ¢ — ¢,, will 


represent the space traversed ; hence 


yvtv 
S—S, = coer cr t,) : (4) 
. Vv UY Sag Z, : 
or, since > = eee we have, in another form, 


s raked meas u,(z ai t,) a WAG sr to)". : (4) 
Multiplying equations (3) and (4), we obtain 
vi=vu, + 27f(s— S,). (5) 


When the particle starts from rest, v, = 0; and if we take the 
starting point as the origin from which to reckon s, and the 
time of starting as the origin of time, then s, = 0, ¢, = 0, and 
equations (3), (4), and (5) become v =/¢, s = $f’, and wv = 2fs. 

Formula (4) may also be obtained by a ‘geometrical con- 
struction. 

At the extremities of a line AB (Fig. 5), equal in length to 
¢ — t,, erect perpendiculars AC and SD, proportional to the 


16 ELEMENTARY PHYSICS. [18 


initial and final velocities of the moving particle. For any in- 
terval of time Aa so short that the veloc- D 
ity during it may be considered constant, 
the space described is represented by the 
rectangle Ca, and the space described in 
the whole time ¢ — ¢,, by a point moving 
with a velocity increasing by successive pqivq 

equal increments, is represented by a Fic. 5. 
series of rectangles, eb, fc, gd, etc., described on equal bases, aé, 
bc,cd,etc. If ab, be... be diminished indefinitely, the sum of 
the areas of the rectangles can be made to approach as nearly 
as we please the area of the quadrilateral ABCD. This area, 
therefore, represents the space traversed by the point, having 
the initial velocity v,, and moving with the acceleration /, 
through the time ¢—7,. But ABCD is equal to AC (¢ —Z,) + 
(BD — AC) (¢ —7¢,) +2; whence 


See v,(¢ ren t) 2 afle Man t,)*. (4) 


B: 


18 Composition and Resolution of Motions, Velocities, 
and Accelerations.—lIf a point a, move witha constant veloc- 
ity relative to another point @,, and this point a, move with a 
constant velocity relative to a third point a,, then the motion, 
in any fixed time, of a, relative to a, may be readily found. 

Represent the motion, ina fixed time, of a, relative to a, (Fig. 
6) by the line v,, and of a, relative to a, by the line v,. Now, 
itis plain that the motions v, and v,, whether acting succes- 
sively or simultaneously, will bring the point a, to B; and also 


gp that if any portions of these motions 
ee | Ab and dc, occurring in any small por- 
d 


[eee tion of time, be taken, they will, be- 
A 7 cause the velocities of a, and a, are con- 
Fic. 6, stant or proportional to v, and v,, bring 


the point a, to some point ¢ lying on the line joining A and 


18] MECHANICS OF MASSES. £7. 


B. Therefore the diagonal AB of the parallelogram having the 
sides v, and v, fully represents the motion of a, relative to a,. 

The line AB is called the resultant, of which the two lines 2, 
and wv, are the components. 

This proposition may now be stated generally. The result- 
ant of any two simultaneous motions, represented by two lines 
drawn from the point of reference, is found by completing the 
parallelogram of which those lines are sides; the diagonal drawn 
from the point of reference represents the resultant motion. 

The resultant of any number of motions may be found by 
obtaining the resultant of any two of the given components, 
by means of the parallelogram as before shown, using this re- 
sultant in combination with another component to obtain a 
new resultant, and proceeding in this way until all the compo- 
nents have been used. 

The same result is reached by laying off the components as 
the consecutive sides of a polygon, when the line required to 
complete the polygon is the resultant sought. 

The components of a given motion in any two given direc- 
tions may be obtained by drawing lines in the two directions 
from one extremity of the line representing the motion, taken 
as origin, and constructing upon those lines the parallelogram 
of which the line representing the motion is the diagonal. The 
sides drawn from the origin represent the component motions 
in direction and amount. 

A motion may be resolved in three directions not in the 
same plane by drawing from the extremity of the line repre- 
senting the motion, taken as origin, lines in the three given direc- 
tions, and constructing upon those lines the parallelopiped of 
which the line representing the motion is the diagonal. The 
sides of the parallelopiped drawn from the origin represent the 
required components. 

Motions are usually resolved along three rectangular axes 
by means of the trigonometrical functions. Thus, if @ be the 

2 


18 ELEMENTARY PHYSICS. [19 


line representing the motion, and 9, ¢, and # the angles which 
it makes with the three axes, the components along those axes 
are a cos 6, a cos ¢, and a cos yp. 

Two motions may be compounded by first resolving them 
along two rectangular axes in their plane, and obtaining the 
resultant of the sums of their components along the axes. If 
a and 6 (Fig. 7) represent motions, @ 
cos ¢, 6 cos 6, a sin d, 6sin 6 are the 
resolved components of a and d along 
the: axes: 

Let a cos@+ bcos 9@=YX and 
asing@+ sin é@= JY; then the diago- 
nal of the rectangle, of which XY and 
Y are-sides, is R = (X’* + Y"}; or, 
since the angle between the resultant 
and the axis of X is known by Y = X& tan 4, it follows that 
Ue rem ee or te It is evident that this process may be 

cos ~ sin J 
extended to any number of components in the same plane. 

It is to be noted that the parailelogram law, though only 
proved for motions, can be shown by similar methods to be 
applicable to the resolution and composition of velocities and 
accelerations. 

19. Simple Harmonic Motion.—If a point move in a 
circle with a constant velocity, the point of intersection of a 
diameter and a perpendicular drawn from the moving point to 
this diameter will have a szzple harmonic motion. Its velocity 
at any instant will be the velocity in the circle resolved at that 
instant parallel to the diameter. The radius of the circle is the 
amplitude of the motion. The period is the time between any 
two successive recurrences of a particular condition of the 
moving-point. The position of a point executing a simple 
harmonic motion can be expressed in terms of the interval of 
time which has elapsed since the point last passed through the 


Fic. 7. 


19] | MECHANICS OF MASSES. 19 


middle of its path in the positive direction. This interval of 
time, when expressed as a fraction of the period, is the phase. 
We further define rotation in the positive direction as that 
rotation in the circle which is contrary to the motion of the 
hands of a clock, or counter-clockwise. Motion from left to 
tight in the diameter is also considered positive. Displace- 
ment to the right of the centre is positive, and to the left 
negative. , Le 
If a point start from X (Fig. 8), the position of greatest 
positive e/ongation, with a simple harmonic motion, its distance 
s from O or its atsplacement at the end of the time ¢, during 
which the point in the circle has 

pee moved through the are BX, is 

Fe Cie OBicos; da)" Now,. Of» is 

B equal. "to © O.\,) the amplitude, 

2nt 
a 


where 7 is the period; hence 


represented by a, and @ = 


2nt 
SSE COB er 6 
7 (6) 
ads To find the velocity at the 
Fic. 8. point C, we must resolve the ve- 
locity of the point moving in the circle into its components 
parallel to the axes. The component at the point C along OX 


; 27a 
feu tein. Or, since V — a 


ROT eas ”) 


remembering that motion from right to left is considered 
negative. 


20 EBLEMEN TATIVAIPHY SIGS: [19 


In order to find the acceleration at the point C directed 
towards O, we must find the rate of change of the velocity at C 
given by Eq. (7). Since, if the point is moving with an accel- 
eration, the velocity increases with the time, as the time in- 
creases by a small increment JZ, the velocity also increases by 
the increment Jv. Eq. (7) then becomes 


2 _ | 2at a At 
odo — 2 on Sl 
27a 7. 2nt he 2a4t a 2nt ia eee 
= — —— {sin —— —— —— si i 
6 ( i ie Tan 
aA woe : 
As 4t approaches zero, cos F approaches the limit unity, 


“ 
~ 


2a At 
i making these 


z s 
and sin can be replaced by its arc 


A 
if 
changes, and transposing, 

Av Ana 2nt 
Ab ae 


a “au dv 
But in the limit where these changes are admissible, ae 


o 


av : ‘ ; 
becomes ane that is, the acceleration of the point. 


Hence the acceleration sought is 


47° 2nt 


f=- Fre COS FR (8), 


This formula shows that the acceleration in a simple har- 
monic motion is proportional to the displacement. It is of the 


19] MECHANICS OF MASSES. 21 


opposite sign from the displacement; that is, acceleration to 
the right of O is negative, and to the left of O positive. 

It is often necessary to reckon time from some other posi 
tion than that of greatest positive elongation. In that case 
the time required for the moving-point to reach its greatest 
positive elongation from that position, or the angle described 
by the corresponding point in the circumference in that time; is 
called the epoch of the new starting-point. In determining the 
epoch, it is necessary to consider, not only the position, but 
the direction of motion, of the moving-point at the instant 
from which time is reckoned. Thus, if Z, corresponding to 
K in the circumference, be taken as the starting-point, the 
epoch is the time required to describe the path LX. But if L 
correspond to the point A’ in the circumference, the motion 
in the diameter is negative, and the epoch is the time required 
for the moving-point to go from Z through O to X’ and back 
to X. 

The epochs in the two cases, expressed in angle, are, in the 
first, the angle measured by the arc AX; and, in the second, 
the angle measured by the arc K'X’K X. 

Choosing X in the circle, or Z in the diameter, as the point 
from which time is to be reckoned, the angle ¢ equals angle 
KOB — angle KOX, or m — e, where ¢ is now the time re- 
quired for the moving-point to describe the arc AS, and é is 
the epoch or the angle KOX. 

The formulas then become 


as 

Si=— @ COS ly gi ae ; 
27 2nt 

CP mms a @sin —— é) 
Am 


22 ELEMENTARY PAYSICS. [1> 


Returning to our first suppositions, letting XY be the point 
from which epoch and time are reckoned, it is plain that, since 


BO aein gi acos(¢ — =) = acos(= — 2), 


the projection of B on the diameter OY also has a simple 
harmonic motion, differing in epoch from that in the diameter 


OX by x It follows immediately that the composition of two 


simple harmonic motions at right angles to one another, hay- 
ing the same amplitude and the same period, and differing in 
epoch bya right angle, will produce a motion in a circle of 
radius @ with a constant velocity. More generally, the co- 
ordinates of a point moving with two simple harmonic mo- 
tions at right angles to one another are 


‘4 =acos(@—e) and y= dcos@. 


If ¢ and ¢’ are commensurable, that is, if @’ = ud, the 
curve is re-entrant. Making this supposition, 


xz =acos@cose+asingsine, and y= bcosnd@. 


Various values may be assigned to a, to #,andtow. Leta 
equal 6 and z equal 1; then | 


z=ycose+(a’— vy’) sin é; 


19] MECHANICS OF MASSES. 23 


from which 
2° —2xy cos e+ 7* cos’ € = a’ sin’ € — 7’ sin’ «, 


or, 


x — 2xy cose + 7" = a’ sin’. 


This becomes, when € = 90°, 2 + 7° = a’, the equation fora 
circle. When € = 0°, it becomes  — y = 0, the equation for 
a straight line through the origin, making an angle of 45° with 
the axis of X. With intermediate values of e¢, it is the equa- 
tion foran ellipse. If we make z = 4, we obtain, as special 
cases of the curve, a parabola and a lemniscate, according as 
€=o0°orgo.. If aand dare unequal, and z = 1, we get, in 
general, an ellipse. 

If a line in which a point is describing a simple harmonic 
motion move uniformly ina direction perpendicular to itself, 
the moving-point will describe a harmonic curve, called also a 
sinusoid. It is a diagram of a semple wave. If the ordinates 
of the curve represent displacements transversely from a fixed 
line, the curve is the diagram of. such waves as those of the 
ether which constitute light. If the ordinates of the curve 
represent displacement longitudinally from points of equilibri- 
um along a fixed line, the curve may be employed to represent 
the waves which occur in the air when transmitting sound. 
The length of the wave is the distance between any two iden- 
tical conditions of points on the line of progress of the wave. 
The amplitude of the wave is the maximum displacement from 
its position of equilibrium of any particle along the line of 
progress. 

If we assume the origin of co-ordinates such that the epoch 
of the simple harmonic motion at the axis of ordinates is 0, 


24. ELEMENTARY PHYSICS. [19 


the displacement from the line of progress of the point describ- 
ing the simple harmonic motion is represented by 


z 
$ = a OS | 27). 
(er7, 


The displacement due to any other simple harmonic motion 
differing from the first only in the epoch is represented by 


z 
Sim LOG (20s — e). 


We shall now show, in the simplest case, the result of com- 
pounding two wave motions. 


The displacement due to both waves is the sum of the dis- 
placements due to each, hence 


t 
Sepas perme | cos 27 + cos (20 4 os é) | 
aie ai pa) j Ks { | 
= @| Cos 24 > + cos 2 F cos € + sin 22 F sine 
t Bi 
=a cos 2m (I + cos €) + sin 27 > Sin e|. 
If for brevity we assume a value A and an angle @ such that 


A cos ¢ = ali + cos 6), 
and 
A sin @?= asin €, 


we may represent the last value of s + s, by 


2 
A cos (anf — ) 


19| MECHANICS OF MASSES. 25 


From the two equations containing A, we obtain, by adding 
the squares of the values of A sin ¢ and A cos ¢, 


A = (2a°-+ 22’ cos e€)}; 


and, by dividing the value of A sin @ by that of A cos ¢, we 
obtain 


sin € 


Ua tian sa, 
a I + coseé 


The displacement thus becomes 


sin € 


z 
s+s, =a(2-+ 200s €)}§ cos (20-5 — tan’ gue mt) (9) 


This equation is of great value in the discussion of prob- 
lems in optics. 

The principle suggested by the result of the above discus- 
sion, that the resultant of the composition of two simple har- 
monic motions is a harmonic motion of which the elements 
depend on those of the components, can be easily seen to 
hold generally. | 

A very important theorem, of which this principle is the 
converse, was given by Fourier. It may be stated as follows: 
Any complex periodic function may be resolved into a number 
of simple harmonic functions of which the periods are com- 
mensurable with that of the original function. 

As an example, any wave not simple may be decomposed 
into a number of simple waves the lengths of which are to each 


26 ELEMENTARY PHYSICS. [20 


other as 4, 4, 4, etc. The number of these simple waves is, in 
general, infinite, but in special cases determinate both as to 
number and to period. 


20. Force.—Whenever any change occurs, or tends to 
occur, in the momentum of a body, we ascribe it to a cause’ 
called a force. 

Whenever motions of matter are effected by our direct 
personal effort, we are conscious, through our muscular sense, 
of a resistance to our effort. The conception of force to which 
this consciousness gives rise, we transfer, by analogy, to the 
interaction of any bodies which is or may be accompanied by 
change of momentum. The question whether this analogy is 
or is not valid, is not involved in a purely physical discussion 
of the subject. A force, in the physical sense, is the assumed 
cause of an observed change of momentum. It is known and 
measured solely by the rate of that change. 

If a body be moving with any acceleration whatever, the 
force acting on it is fully expressed by the product of the mass 
of the body into its acceleration. 

The formula for force is, therefore, 


(Ld BU 
eats — ttf. (10) 
iow 


The dimensions of force are ZL 7-?. 


As acceleration is always referred to some fixed direction, 
it follows that force is a quantity having direction. 

The product of the time during which a force acts by its 
mean intensity is called the zwpulse of the force. 

The practical unit of force is the dye, which is the force 
that can impart to a gram of matter one unit of acceleration; 
that is to say, one unit of velocity in one second. 


22] MECHANICS OF MASSES. 27 


21. Field of Force.—A field of force is a region such that 
a particle constituting a part of a mutually interacting system, 
placed at any point in the region, will be acted on by a force, 
and will move, if free to do-so, in the direction of the force. 
The particle so moving would, if it had no inertia, describe 
what is called a “ine of force, the tangent to which, at any 
point, is the direction of the force at that point. The strength 
of field at a point is measured by the force developed by unit 
quantity at that point, and is expressible, in terms of lines of 
force, by the convention that each hne represents a unit of 
force, and that the force acting on unit quantity at any point 
varies as the number of lines of force which pass perpendicu- 
larly through unit area at that point. Each line, therefore, 
represents the direction of the force, and the number of lines 
passing through unit area, the strength of field. An assem- 
blage of such lines of force considered with reference to their 
bounding surface is called a tube of force. 

22. Newton’s Laws of Motion.—We are now ready for 
the consideration of the laws of motion, first formally enun- 
ciated and successfully applied by Newton, and hence known 
by his name: | 

LEX I.—Corpus omne perseverare in statu suo quiescendi 
vel movendi uniformiter in directum, nisi quatenus illud a 
viribus impressis cogitur statum suum mutare. 

LEx II.—Mutationem motus proportionalem esse vi mo- 
trici impressae & fieri secundum lineam rectam qua vis illa 
imprimitur. : 

Lex III.—Actioni contrariam semper & aequalem esse 
reactionem; sive corporum duorum actiones in se mutuo 
semper esse aequales & in partes contrarias dirigi. 

The subjoined translations are given by Thomson and Tait: 

Law I.—Every body continues in its state of rest or of 
motion in a straight line, except in so far as it may be com- 
pelled by force to change that state. 


28 ELEMENTARY LAY SICS. [23 


Law II.—Change of motion is proportional to force ap- 
plied, and takes place in the direction of the straight line in 
which the force acts. 

Law III.—To every action there is always an equal and 
contrary reaction: or, the mutual actions of any two bodies 
are always equal and oppositely directed. 

23. Discussion of the Laws of Motion.—(1) The first 
law is a statement of the important truths implied in our defi- 
nition of force,—that motion, as well as rest, is a natural state 
of matter; that moving bodies, when entirely free to move, 
proceed in straight lines, and describe equal spaces in equal 
times; and that force is the cause of any deviation from this 
uniform rectilinear motion. 

That a body at rest should continue indefinitely in that 
state seems perfectly obvious as soon as the proposition is 
entertained; but that a body in motion should continue to 
move in a straight line is not so obvious, since motions with 
which we are familiar are frequently arrested or altered by 
causes not at once apparent. This important truth, which is 
forced upon us by observation and experience, may, however, 
be presented so as to appear almost self-evident. If we con- 
ceive of a body moving in empty space, we can think of no 
reason why it should alter its path or its rate of motion in any 
way whatever. 

(2) The second law presents, first, the proposition on 
which the measurement of force depends; and, secondly, 
states the identity of the direction of the change of motion 
with the direction of the force. Motion is here synonymous 
with momentum as before defined. The first proposition we 
have already employed in deriving the formula representing 
force. The second, with the further statement that more 
‘than one force can act on a body at the same time, leads 
directly to a most important deduction respecting the com- 


24] MECHANICS OF MASSES. 29 


bination of forces; for the parallelogram law for the resolution 
and composition of motions being proved, and forces being 
proportional to and in the same direction as the motions 
which they cause, it follows, if any number of forces acting 
simultaneously on a body be represented in direction and 
amount by lines, that their resultant can be found by the 
same parallelogram construction as that which serves to find 
the resultant motion. This construction is called the paral- 
lelogram of forces. 

In case the resultant of the forces acting on a body be 
zero, the body is said to be in equzlibrium. 

(3) When two bodies interact so as to produce, or tend to 
produce, motion, their mutual action is called a stress. If one 
body be conceived as acting, and the other as being acted on, 
the stress, regarded as tending to produce motion in the body 
acted on, isa force. “The third law states that all interaction 
of bodies is of the nature of stress, and that the two forces 
constituting the stress are equal and oppositely directed. 

From this follows directly the deduction, that the total 
momentum of a system is unchanged by the interaction of its 
parts; that is, the momentum gained by one part is counter- 
balanced by the momentum lost by the others. This princi- 
ple is known as the conservation of momentum. 

24. Collision of Bodies.—If two bodies, m, and m,, with 
velocities v, and v, in the same line, impinge, their velocities 
after contact are found, in two extreme cases, as follows: 

(1) If the bodies are perfectly inelastic, there is no tend- 
ency for them to separate, their final velocities will be equal, 
and their momentum will be equal to the sum of their sepa- 
rate momenta; hence 


m,V, + mV, = (m,-+ m,)x, (11) 


where ~ is the velocity after impact. 


30 ELEMENTARY PHYSICS. [25 


(2) If the bodies are perfectly elastic, they separate with a 
force equal to that by which they are compressed. 

Let vw represent their common velocity just at the instant 
when the resistance to compression balances the impulsive 
force. Then the change in momentum in each body up to 
this instant is #,(v, —v), or m,(v — v,); and the further change 
of momentum, by reason of the elasticity of the bodies, is the 
same; whence the whole momentum lost by the one is 
2m(v,—v) and that gained by the other 2m,(v—v,). Ifz 
represent the final velocity of m, we have the equation 


M,V, — M1,x = 2m,v, — 2m,v, 
whence 
we ee 2UV ate Vy. 


In like manner, if y represent the final velocity of m,, we find 
JV = 2U — V,z. 


From the formula for inelastic bodies, which is applicable at 
the moment when both bodies are moving with the same 
velocity, 
m1,U, + ,V, 
tt, +- My? 


whence, finally, 


__ (mM, — m,)v,-+ 2m,0, 
ny. mn, + Ut, 


(mm, — 1,0, -F 2,0, 
big m, + mM, ; 


(12) 


25. Inertia.— The principle of equality of action and re- 
action holds equally well when we consider a single body as 


26] MECHANICS OF MASSES. 31 


acted on by a force. The resistance to change of motion 
offered by the inertia of the body is equal in amount and 
opposite in direction to the acting force. Inertia is not of 
itself a force, but the property of a body, enabling it to offer 
a resistance to a change of motion. 

26. Work and Energy.—When a force causes motion 
through a space, it is said to do work. 

The measure of work is the product of the force and the 
space traversed by the body on which the force acts. The 
formula expressing work is therefore 


mfs. (13) 


The dimensions of work are WL’*T~’. 

In the defined sense of the term, no work is done upon a 
body by a force unless it is accompanied by a change of posi- 
tion, and the amount of work is independent of the time 
taken to perform it. Both of these statements need to be 
made, because of a natural tendency to confound work with 
conscious effort,and to estimate it by the effect on our system. 

A body may, in consequence of its motion or position with 
respect to other bodies, have a certain capacity for doing work. 
This capacity for doing work is its exergy. Energy is of two 
kinds, usually distinguished as fotential and kinetic. The 
former is due to the position of the body, the latter to its 
motion. 

Since the potential energy of a body is due to the exist- 
ence of a force acting upon it, it is clear that, if the body be 
free to move, it will be moved by the force, and its potential 
energy will be diminished. Hence, in any system of bodies 
free to move, movements will occur until the potential energy 
of the system becomes a minimum. 

If a mass m be moving with a velocity v, its capacity for 
doing work may be determined from the consideration, that, 


32 ELEMENTARY PHYSICS. [27 


if the motion be opposed by a force # equal to mf, the mass 
suffers a negative acceleration f, and is finally brought to rest 


after traversing a space s in opposition to the force. From 
Eq. (5) we have s = oF Multiplying both sides of this equa- 
mv 


tion by #= m/f, we have fs = -— 


But Fs is the work done 


by the body against the force /, and is, therefore, the capacity 

which the body originally had for doing work. This capacity 

—that is, the kinetic energy of the body—is then represented 
ke Aes 

by the expressiot Ser 

The dimensions of energy are J7L*7 —’, the same as those 
of work. Since the square of a length cannot involve direction, 
it follows that energy is a quantity independent of direction. 

The practical unit of work and energy is the erg. 

It is the work done against a force of one dyne, in moving 
its point of application in the line of the force through a space 
of one centimetre; 

Or, it is the energy of a body so conditioned that it can 
exert the force of one dyne through a space of one centimetre ; 

Or, it is the energy of a mass of two grams moving with 
unit velocity. 

27. Conservation of Energy.—The difference between 
the kinetic energy of a body at the beginning and at the end 
- of any given path is equal to the work done in traversing that 
path. For, if we consider the mass # having an acceleration 
jf, and moving through a space s,so small that the acceleration 
may be assumed constant, we have, from Eq. (5), 


v=v, + 2fs, 


where s replaces the s — s, of the equation. 


27] MECHANICS OF MASSES. 33 


Multiplying by $, we have 


ymv' =4mv, + mfs. 


Since any motion whatever may be divided into portions in 
which the above conditions hold true, it follows that we have 
finally, for any motion, 


mv = 3m, + mfp s, t+ mfs,...= 4m, + 2mfs. 


Since $mv,’, or the initial kinetic energy, is a constant 
quantity, it follows that $mv*— 2mfs, or the sum of the 
kinetic and potential energies, is a constant quantity for any 
body moving under the action of forces without collision with 
other bodies. In- other words, a body, by losing potential 
energy, gains an equal amount of kinetic energy; and the 
kinetic energy, being used to do work against acceleration, 
places the body in a position where it again possesses its 
original amount of potential energy. 

This statement holds true for any body of a system made 
up of bodies moving, without collision, only under their mu- 
tual interactions. It follows therefore that the total energy of 
such a system remains constant. 

There are other forms of energy besides the potential and 
kinetic energies of masses. By suitable operations energy in 
any one form may be transformed into energy of any other 
form. The simplest example of such a transformation is the 
simultaneous production of heat and loss of mechanical en- 
ergy by friction or collision. 

In any closed system, into which no energy enters, and out 
of which no energy passes, the statement made above for the 
energy of a simple system of bodies holds true, if all forms of 

3 


nA ELEMENTARY <P iT¥ SICS. [28 


energy in the system are taken into account. Whatever trans- 
formations of the energy within the system occur, its totai 
amount remains constant. ‘This principle, called the principle 
of the conservation of energy, can be demonstrated to hold for 
the mechanical interaction of bodies moving without collision, 
and has been established by experiment for operations involv- 
ing molecular and atomic interactions. It is a general prin- 
ciple, with which all known laws of the material universe are 
consistent. 

The principle of eve conservation of energy is so well estab- 
lished and so universally accepted, that, where convenient, it 
has been used in the demonstrations of this book as a funda- 
mental principle. 

28. Difference of Potential—The azfference of potential 
between two points in a field of force is measured by the work 
done by the forces of the field in moving atest unit of the 
quantity to the presence of which the force is due from one 
point to the other. : 

If Vp— Vo represent the difference of potential pete es, 
the points Pand Q, and if / represent the average force be- 
tween those points and s the distance between them, then the 
amount of work done in moving a unit from P to Q, and 
hence the difference of potential between P and Q, is rep- 
resented by 


Vp == Vo mn gS 
From this relation we have 


op Sb Cero 20 le omeeier 

s s 
If s become indefinitely small, in the limit / represents the 
Vo— Vp aV 


-—— = — — becomes the 
S as 


force at the point P, and — 


28] MECHANICS OF MASSES. 35 


rate of change of potential at that point with respect to space, 
taken with the opposite sign. Hence we obtain a definition of 
potential. It isa function, the rate of change of which at any 
point, with respect to space, taken with the opposite sign, 
measures the force at that point. 

In the discussion which follows we deal with forces which 
vary directly as the product of the quantities acting, and in- 
versely as the squares of the distances which separate them. 
For convenience, these acting quantities will be called masses 
or quantities of matter. By the substitution of proper terms 
the theorems to be presented will hold equally well in all cases 
involving forces acting according to this law. 

If the field be due to the presence of a mass #, which repels 
the test unit, at a distance s, with a force expressed byy 


the difference of potential between two points, distant 7 and 
R from the mass m, is expressed by 


I I 
Pe Vp im( — aT 


The symbol £ represents the force with which two unit masses 
at unit distance repel one another. 

To obtain this formula in the simplest case, let us suppose 
a mass at the point O (Fig. 9) acting upon 


3 O Ls ROP 
; ; C777 ee ee eee Le 
aunit at Pwith a force equal to Op If Hien 


. km 
the unit be moved to Q the force at Q is OO! and the average 
force acting while the unit is moving in the path PQ, provided 


36 ELEMENTARY PHYSICS. [28 


this path be taken small enough, is 


kim 


OP- OQ" 
The work done in moving the unit through PQ is . 


(apap)? be oP ao? Asi 6Y0)\ = anlar bb aa 


The work done in moving the unit through any other small 
space OR towards S is, similarly, 


kim Gen, — ao) ; 


The last value obtained by moving the unit from S to TJ is 


I I 
in| aa, — oe 


The sum of these values, 


in op ap) 


gives the work done in moving the unit through the space PT. 

It is evident that the amount of work done upon the 
unit to move it from P to J is independent of the path. 
For, if this were not so, it would be possible by moving from 
Pto Ton one path, and returning from 7 to Pon another, to 
accumulate an indefinite amount of energy ; which the principle 
of the conservation of energy shows to be impossible. 


28] MECHANICS OF MASSES. ay 


Since the point Z can be considered as on the surface of a 
: : kum 

sphere of which O is the centre, and since the force of? acts 
along 7O perpendicular to that surface, and cannot, therefore, 
have a component tending to produce motion on that surface, 
there is no work done in moving the unit at 7 over the surface 
of the sphere to any other point X on it: it follows that the 
difference of potential between P and any other point at dis- 


tance v from O is the same as that between P and 7. Whence 


V,—Vr= am(t a “) (14) 


Such a surface as the one described, to which the lines of 
force are perpendicular, is called an equzpotential surface. 

If the point P be supposed to be at a distance from O so 
great that the force at that distance vanishes, it is then at zero 
potential. R becomes indefinitely large, and the absolute poten- 
tial at T becomes 


Mme + (15) 


This formula expresses the work necessary to move the 
unit against the repulsion of the mass at O up to the point 7 
from an infinite distance. If the mass attract the unit, the 
work is done by the attraction upon the unit in so moving up 
to 7, and the potential is negative. 

From the definitions, it is plain that the difference of poten- 
tial between P and 7 equals the difference of the potential 
energies of a unit at those points. 

If the potential of any point be due to the action of more 
than one mass, it is found by adding the potentials due to the 


38 ELEMENTARY PHYSICS. [29 


separate masses. If 2 be a summation sign indicating this 
operation, Eqs. (14) and (15) become 


a oe 
Vp —Vo= Shm\t — a, (14) 
and 
V, ae pis (15) 


29. Theorems relating to Difference of Potential.— 
(1) The force at any point within a spherical shell of uniform 
thickness and density is zero. For, if 6 represent the density 
of the shell, and if ad, and cd, represent 
the volumes of the portions of the 
shell cut out by a cone having its apex 
at O (Fig. 10), then the force, if an attrac- 


; i . kad, 
tion, acting towards a is —, and towards 
! Oa 


. Roca, 
cis Oe: Hence the efficient force tend- 


ing to produce motion, say towards a, is expressed by 


ab, fel) 
sete Eee 

Now, if a0,, cd,, be taken small enough, they will be frusta 
of similar cones, and, as a consequence, 


29] MECHANICS OF MASSES. 39 


from which, since the density of the shell is uniform, 


ab, cd, 
BG — Oe) 
Since the whole surface of the shell may be cut by similar 
cones, for which similar equations will hold, the total force ex- 
erted by the shell on a unit within it becomes zero. This be- 
‘ing so, it follows that the potential throughout the sphere is 
constant; for no work is required to move the unit from one 
point to another in the interior. 
(2) The potential, and therefore the force at a point, due to 
the presence of a spherical shell of uniform density, depends 


K 


a= 
eZ 


FiGuixr, 


B 


only on the mass of the shell and on the distance of the 
point considered from the centre of the sphere. Let CKL 
(Fig. 11) represent a central section of the shell, of which O 
is the centre. Let d represent the mass of a portion of the 
shell having unit of area. The potential at 2, due to the 
element of the sphere at A, having an area represented by s, is 
i me ae and the potential due to the whole sphere is the 


summation of that due to all the similar elements making up 


40 ELEMENTARY PHYSICS: [29 


the sphere. Take a point 4 onthe line OP, such that OA .OB 
= k’, where & is the radius of the sphere; draw AKA, produce 
it to Z, and draw OK and OL. Now, if we represent the angle 
OKA by a, and the solid angle subtended by the element s, as 


AK?’ .@ 
seen from A, by #, we may express s in other terms as cout ; 
hence 

iv aA Ko 
LORRI CaR me 


Now, since, by construction, OB: R= R: OA, and the 
angle KOB is common to the two triangles KOA and BOK, 
these triangles are similar; hence 


AK R 


BK = 08 
The value of the potential due to s may then be written 


TG Otte 
pions 


er Saiaee Gama B @). 
‘ cosa OB 


The value for the potential of the corresponding element 


at L is, similarly, ‘ 


| 2B Bits 
Vu = eos a OB 


Adding these values, we obtain 


VT Vig ae 


OB cose 


R pees =), 


29| MECHANICS OF MASSES. 4! 


But 
: 6 
AK + AL a ALT — 20K = 2k; 
cos @ cos @ 


hence we obtain, finally, 


R’-@ 


V+ V,, = 24a 


Now the sphere may be divided into two portions, made 
up of elements similar to A and ZL, by a plane passing through 
A normal to OS. We obtain the whole potential, therefore, 
by summing all the potential values due to these pairs of 
elements; whence 


R? 
[Si 2d DH R= @: 


The sum of all the elementary solid angles on one side of 
the plane from the point A in it is 27; hence, finally, 


v= Rd iy oe 
mTOR = OF: 


where m is the mass of the spherical shell. 

Since the force at the point B depends on the rate of change 
of potential at that point with respect to space, it varies in- 
versely as the square of the distance OB. Represent OB by Z. 


In the expression bee et] alin by asmall increment 47, 


“5 ? 
and denote the corresponding change in the potential by JV; 
then 

Mm 


V+ AV =73-ap 
Vit VAI4I4V+4VMA=m 


42 ELEMENTARY PHYSICS. [29 


If 47 become indefinitely small, in the limit the product 
AVAl may be neglected. We then have 


V4it l4V=o, 
or 


AR het A Ai Saipan, | 6 mM 
TD Withee] ee) Sey aa a 

This is the rate of change of potential at the point 2, with 
respect to space, and, taken with the opposite sign, measures 
the force at that point. The force, therefore, at a point out- 
side a spherical shell of uniform density varies inversely as the 
square of its distance from the centre of the sphere. This 
‘result enables us to deal with spheres of gravitating matter, 
or spherical shells, upon which is a uniform distribution of 
electricity, as if they were gravitating or electrified points. 

(3) If in the last proposition we let 7= &, we obtain for 
the value of the force just outside the shell 


seis —— Amd. 

Since the force just inside the shell vanishes, in conse- 
quence, as we have seen, of the equal and opposite actions of 
the portions of the sphere ad and acd (Fig. 12), and since the 
total force at the point P outside the sphere 
is 47d, it follows that the force at P, due to 
ab, is 27d. If the radius be taken large 
enough, av may be considered as flat, and 
constituting a disk: hence the force at a 
point near a flat disk of density d is 27d. 
Since the force at a point near one surface 


Fic, 12. 
of the disk is 27d in one direction, and near the other surface 


30] MECHANICS OF MASSES. 43 


27d in the other direction, it is clear that, in passing through 
the disk, the force changes by a7d. 

30. Moment of Force.—The moment of force about a point 
is defined as the product of the force and the perpendicular 
drawn from the point upon the line of direction of the force. 

The moment of a force, with respect to a point, measures 
the value of the force in producing rotation about that point. 

If momentum be substituted for force in the foregoing defi- 
nition, we obtain the definition of szoment of momentum. 

In order to show that the moment of a force measures the 
value of that force in producing rotation, we will find the di- 
rection and amount of the resultant of two forces in the same 
plane acting on a rigid bar, but not applied at the same point. 

Let BD (Fig. 13) be the bar, DF and BG the forces. Their 
lines of direction will, in general, meet at some point as O. 
Moving the forces up to O, and applying the parallelogram of 
forces, we obtain the resultant O/, 
which cuts the bar at A. If we 
resolve both forces separately, 
parailel to O/7 and SD, this re- 
sultant equals in amount the sum D 
of those components taken paral- 
lel to O/7. Hence the compon- 
ents EF and CG, taken parallel to F E Fig, x3. 

DB, annul one another’s action, and, being in opposite direc- 
tions, are equal. Now, by similarity of triangles, 


CA AL = bOCE 
and 
OA AD= DES Er: 


whence, since CG = EF, we obtain 


AB* BG anes 


44 ELEMENTARY PHYSICS. [31 


Resolving both DZ and BSC perpendicular to Db, we see that 
the moments of force about A are equal. Now, if the result- 
ant O/ be antagonized by an equal and opposite force applied 
at A, there will be no motion. Hence the tendencies to rota- 
tion due to the forces are equal,—a result which is in accord 
with our statement that the moment of force is a measure of 
the value of the force in producing rotation. 

The resultant of two forces may be found in general by 
this method. The case of most importance is the one in 
which the two forces are parallel. The lines DE and SC in 
the diagram represent such forces. It is plain, from the dis- 
cussion, that these forces also will have the force represented by 
O/ as their resultant, applied at the point A. The resultant 
of two parallel forces applied at the ends of a rigid bar is then 
a force equal to their sum applied at a point such that the two 
moments of force about it are equal. 

31. Couple—The combination of two forces, equal and 
oppositely directed, acting on the ends of a rigid bar, is called 
a couple. By the preceding proposition, the resultant of these 
forces vanishes, and the action of a couple does not give rise 
to any motion of translation. The forces, however, conspire 
to produce rotation about the mid-point of the bar. It follows 
from the fact that a couple has no resultant, that it cannot be 
balanced by any single force. 

32. Moment of Couple.—The moment of couple is the pro- 
duct of either of the two forces into the perpendicular distance 
between them. It follows from what has been already proved, 
that this measures the value of the couple as respects rota- 
tion. 

33. Centre of Inertia.—If we consider any system of equal 
material particles, the point of which the distance from any 
plane whatever,is equal to the average distance of the several 
particles from that plane,is called the centre of tnertia. This 
point is perfectly definite for any system of particles. It fol- 


33] MECHANICS OF MASSES. 45, 


lows from the definition, that, if any plane pass through the 
centre of inertia, the sum of the distances of the particles on 
one side of the plane, from the plane, will be equal to the sum 
of the distances of the particles on the other side: hence, if 
the particles are all moving with a common velocity parallel to 
the plane, the sum of the moments of momentum on the one 
side is equal to the sum of the moments of momentum on the 
other side. And, further, if the particles all have a common 
acceleration, or are each acted on by equal and similarly di- 
rected forces, the sum of the moments of force on the one side 
is equal to the sum of the moments of force on the other side. 

If we combine the forces acting on two of the: particles, one 
on each side of the plane, we obtain a resultant equal to their 
sum, the distance of which from the plane is determined by the 
distances of the two particles from the plane. Combining this 
resultant with the force on another particle, we obtain a second 
resultant; and, by continuing this process until all the forces 
have been combined, we obtain a final resultant, equal to the 
sum of all the forces, lying in the plane, and passing through 
the centre of inertia. This resultant expresses, in amount, 
direction, and point of application, the force which, acting on 
a mass equal to the sum of all the particles, situated at the 
centre of inertia, would impart the same acceleration to it as 
the conjoined action of all the separate forces on the separate 
particles imparts to the system. When the force acting is the 
force of gravity, the centre of inertia is usually called the centre 
of gravity. | | 

When the forces do not act in parallel lines, the proposi- 
tion just stated does not hold true, except in special cases. 
Bodies in which it still holds are, for that reason, called cenxtro- 
baric bodies. . 

The centre of inertia can be readily found in most of the 
simple geometrical figures. For the sphere, ellipsoid of revolu- 
tion, or parallelopiped, it evidently coincides with the centre of 


46 ELEMENTARY PHYSICS. [34 


figure; sincea plane passing through that point in each case 
cuts the solid symmetrically. 

34. Mechanical Powers.—The preceding definitions and 
propositions find their most elementary application in the so- 
called mechanical powers. 

These are all designed. to enable us, by the application of a 
certain force at one point, to obtain at another point a force, 
in general not equal to the one applied. Six mechanical 
powers are usually enumerated,—the lever, pulley, wheel and 
axle, inclined plane, wedge, and screw. 

(1) Zhe Lever is any rigid bar, of which the weight may be 
neglected, resting on a fixed point called a fulcrum. From 
the proposition in § 30, it may be seen, that, if forces be ap- 
plied to the ends of the lever, there will be equilibrium when 
the resultant passes through the fulcrum. In that case the 
moments of force about the fulcrum are equal; whence, if the 
forces act in parallel lines, it follows that the force at one end 
is to the force at the other end in the inverse ratio of the 
lengths of their respective lever-arms. If Zand Z represent the 
lengths of the arms of the lever, and Pand P the forces ap- 
plied to their respective) extremities, then //—-7-7e 

The principle of the equality of action and reaction enables 
us to substitute for the fulcrum a force equal to the resultant 

of the two forces. We have then a 

combination of forces as represented 

in the diagram (Fig. 14). Plainly any 

P one of these forces may be considered 

Hig 54. as taking the place of the fulcrum, and 
either of the others the power or the wezghi. 

The lever is said to be of the first kind if A is fulcrum and 
P power, of the second kind if P is fulcrum and P power, of 
the third kind if Pis fulcrum and & power. 

(2) The Pulley is a frictionless wheel, in the groove of which 
runs a perfectly flexible, inextensible cord. 


R 


F 


34| MECHANICS OF MASSES. 47 


If the wheel be on a fixed axis, the pulley merely changes 
the direction of the force applied at one end of the cord. If 
the wheel be movable and one end of the cord fixed, and a 
force be applied to the other end parallel to the direction of 
the first part of the cord, the force acting on the pulley is 
double the force applied: for the stress on the cord gives rise 
to a force in each branch of it equal to the applied force; each 
of these forces acts on the wheel, and, since the radii of the 
wheel are equal, the resultant of these two forces is a ferce 
equal to their sum applied at the centre of the wheel. From 
these facts the relation of the applied force to the force ob- 
tained in any combination of pulleys is evident. 

(3) Zhe Inclined Plane is any frictionless surface, making an 
angle with the line of direction of the force applied at a point 
upon it. Resolving the force P (Fig. 15), making an angle @ 
with the normal to the plane, into its com- 
ponents Pcos @ and Psin ¢@ perpendicular 
to and parallel with the plane, Psin ¢ is 
alone effective to produce motion. Con- 
sequently, a force Psin # acting parallel 
to the surface will balance a force P, mak- 
ing an angle @ with the normal to the surface. If the plane 
be taken as the hypothenuse of a right-angled triangle ABC, 
of which the base AB is perpendicular to the line of direction 
of the force, then, by similarity of triangles, the angle BAC 
equals @: whence the force obtained parallel to AC is equal 
to the force applied multiplied by the sine of the angle of in- 
clination of the plane. If the components of the force applied 
be taken, the one, as before, perpendicular to the plane AC, 
and the other parallel to the base AAS, the force obtained 
parallel to AB is equal to the force applied multiplied by the 
tangent of the angle of inclination of the plane. 

(4) The Wheel and Axie is essentially a continuously acting 
lever. 


BiG. rs. 


48 ELEMENTARY PHYSICS. [35 


i a 


(5) Zhe Wedge is made up of two similar inclined planes set 
together, base to base. 

(6) Zhe Screw is a combination of the lever and the in- 
clined plane. 

The special formulas expressing the relations of the force 
applied to the force obtained by the use of these combinations, 
are deduced from those for the more elementary mechanical 
powers. | 
It may be seen, in general, in the use of the mechanical 
powers, that the force applied is not equal to the force ob- 
tained. A little consideration will show, however, that the 
energy expended is always equal to the work done. 

Any arrangement of the mechanical powers, designed to 
-do work, is called a machine. ‘The more nearly the value of 
the work done approaches that of the energy expended, the 
more closely the machine approaches perfection. The elastic- 
ity of the materials we are compelled to employ, friction, and 
other causes which modify the conditions required by theory, 
make the attainment of such perfection impossible. 

The ratio of the useful work done to the energy expended 
is called the effictency of the machine. Since in every actual 
machine there is a loss of energy in the transmission. the effi- 
ciency is always a proper fraction. 

35. Angular Velocity.—The angle contained by the line 
passing through two points, one of which is in motion, and 
any assumed line passing through the fixed point, will, in gen- 
eral, vary. The rate of its change is called the angular velocity 
of the moving-point. If @and ¢, represent the angles made 
by the moving line with the fixed line at the instants ¢ and Zz, 
then the angular velocity, if constant, is measured by 


o— ¢, 
Tee ed On 


If variable, it is measured by the limit of the same expression, 


35] MECHANICS OF MASSES. 49 


— 


ap P io PD, ° ° - 
> aa as the interval ¢ — ¢, becomes indefinitely 
small. 


The angular acceleration is the rate of change of angular 
velocity. If constant, it is measured by 


@) GQ), 
eg eae by (17) 


If variable, it is measured.by the limit of the same expression, 
BD) GO, 
Ra okies, 
small. 

If the radian be taken as the unit of angle, the dimensions 


of angle become 
| arc | a L wa 
Pacis len Lake g S 


Hence the dimensions of angular velocity are 7 —*, and of an- 
etiavacceistation, 7 7". 

If any point be revolving about a fixed point asa centre, 
its velocity in the circle varies as its angular velocity and the 
length of the radius jointly. 

Angular velocities may be compounded by a process similar 
to that employed for the composition 
of motions. 

Let OA and OB (Fig. 16) represent 
two axes of rotation about which 
points are revolving with angular ve- 
locities w, and w, respectively; both 
rotations being clockwise when seen : 
from the point O. The velocity at a oY Faerize. 
point Z, at unit distance from O, due to the motion about OA, 

4 + 


as the interval ¢ — ¢, becomes indefinitely 


50 ELEMENTARY PHYSICS. [35 


is @, sin a, and that due to motion about OZ is », sin f in the 
opposite direction. The whole velocity of Z is, therefore, 


ao, sin a — o, sin fi. 


There must be some position of Z for which this velocity be- 
comes zero. Then w, sina = @,sin 6. It follows at once 
that every point on the line OZ is at rest. If we consider OL 
as the axis of rotation, and suppose the angular velocity of 
every point of the system about this axis to be w, such that 
@sina = o, sin (a + f), this angular velocity will give the 
actual velocity of any point. To illustrate by a simple exam- 
ple, we will show that 


sin (a +f) 


sin @ sin § 


asin f = @, 


is the velocity at B at unit distance from O. The velocity at 
& is only due to rotation about OA, and is therefore given by 
w, sin (a+ f), From our previous equation, 


@, sin a = @, sin f; 


hence 
cw, sin (a + p) = @, OTP) cin , 

Now 
gan nee pula ie 


sin @ 


and the equality of the expressions is shown. Similarly it may 
be shown that the value of the velocity at any point WV at 
unit distance from O, as given by the expression o@ sin VOZ, is 


equal to that given by w, sin dON— @, sin BON. The twe 


35] MECHANICS OF MASSES. 51 


rotations about OA and OF may thus be combined to form 
one rotation about OZ. 

Draw LF and LE parallel to OA and OS. Then the lines 
O£ and OF are numerically proportional to the angular veloci- 
ties w, and w,; for, since OFLF is a parallelogram, 


sin a OF 
sing OF 


But, from our first equation, 


whence 
ORO Hiss @ae Gl 


The line OZ is likewise proportional to o, for, from the 
figure, . 


Qin i= sit asin. (a+. 6) ; 


whence we see immediately, from the equation giving the value 
ef w, that 


OL: 0F = 0:0, 


We can therefore obtain the direction of the resultant axis, 
and the amount of the angular velocity, due to rotation about 
two other axes, by laying off on those axes, from their point 
of intersection, lengths numerically equal to the angular veloci- 
ties about them, and drawing the diagonal of the parallelogram 
of which they are the sides. And so also any angular velocity 
may be resolved into three, the axes of which are at right angles 
to one another, by employing the trigonometrical functions of 
the angles which its axis makes with the three component axes. 


52 ELEMENTARY PHYSICS. [35 


It has been demonstrated that if a body be established in 
rotation, for any finite time, about an axis fixed with reference 
to points in it, however the position of the body be altered, 
it will continue to rotate with constant angular velocity about 
the same axis, unless constrained by outside forces to change its. 
rotation. In other words, the axis of rotation always remains 
parallel to itself. This property is of importance in the discus- 
sion of some interesting applications of the preceding princi- 
ples, which we shall next consider. ; 

(1) The first of these is the method employed by Foucault 
to determine by experiment the fact of the earth’s rotation. 
His apparatus consisted of a spherical pendulum bob, sus- 
pended by a truly cylindrical wire, so that it could swing freely 
in any plane. It can easily be seen, that, if such a pendulum 
were set up at the pole and swung, it would preserve its plane 
of oscillation invariable, and the earth would turn around un- 
der it, so that in twenty-four hours the pendulum would seem 
to have traversed a complete circle in the direction of the sun’s. 
apparent motion. At any other point on the earth’s surface 
N the change in apparent direction of the 
plane of oscillation would not be so great. 
Let @ represent the angular velocity of | 
the earth, ¢ the duration of the experi- 
ment, and @ the latitude. Let the pen- 
dulum be supposed to be at 4 (Fig. 17). 
Let VS be the earth’s axis. Now, the 
angular velocity o, represented by OC, 

5 may be resolved into two components, 

sini OD and DC, the axes of which lie respec- 

tively in the direction of the force acting on the pendulum 
and at right angles to it. The angular velocity DC has no. 
influence in changing the relations of the pendulum and the 
earth; but the angular velocity OD = OC sin 6= w sin @ is: 
made evident by the rotation of a fixed line on the earth’s sur- 


A 


35] MECHANICS OF MASSES. 53 


face, cutting the invariable plane of oscillation at the point of 
equilibrium of the pendulum. The plane of oscillation of the 
pendulum consequently appears to rotate in the opposite di- 
rection with an angular velocity w sin ¢, and the angle swept 
out in any time ¢ is w¢ sin @. By such an apparatus has been 
determined, not only the fact of the earth’s rotation, but even 
an approximate value of the length of the day. 

(2) The phenomena presented by the gyroscope also offer 
an example of the application of the foregoing principles. 

The construction of the apparatus can best be understood 


Fig, 3x8. 


by the help of the diagram (Fig. 18). The outermost ring rests 
in aitame, and turns on the points a4,a@, The inner rests in 
“the outer one, and turns on the pivots 4, 0, at right angles to 
the line of aa. Within this ring is mounted the wheel G, the 
axle of which is at right angles to the line 00,, and in a plane 
passing through aa,. At the point ¢ is fixed a hook, from which 
weights may be hung. It is evident that if the wheel be 
mounted on the middle of the axle, the equilibrium of the ap- 
paratus is neutral in any position, and that a weight hung on 
the hook e will bring the axle of the wheel vertical, without 
moving the outerring. If, however, the wheel be set in rapid 


54 ELEMENTARY PHYSICS. [35. 


rotation, with its axle horizontal, and a weight be hung on the 
hook, the whole system will revolve with a constant angular 
velocity about the points a, a, and the axle of the wheel will 
remain horizontal. 

The explanation of this phenomenon follows from the 
principles which we have already discussed. The conditions 
given are, that a body rotating with an angular velocity in one 
plane is acted on by a force tending to produce rotation in a 
perpendicular plane. 

Let the plane of the paper represent the horizontal plane, 
and the line AB (Fig. 19) represent the 
direction of the axle at any mgment. 
Lay off on OA a length OP proportional 
to the angular velocity of the wheel. If 
Pbe the point of application of the weight, 
the weight tends to turn the system about 
an axis CD at right angles to AB. Let 
us suppose, first, that, in the small inter- 
val of time ¢, the system acquires an an- 
gular velocity about CD proportional to 
OQ. Compounding the two angular velocities OP and OQ, we 
obtain the resultant OR. Now, resolving O@ parallel and at 
right angles to OR, we see that the parallel component is effi- 
cient in determining the length of CR, the component at right 
angles, the direction of OR. Inthe limit, as ¢ becomes indefi- 
nitely small, OQ also becomes indefinitely small, and the re- 
solved component Oy parallel to OR vanishes in comparison 
OQ Ox 
ORMpae 
effect will be a change of direction of the axle AZ in the hori- 
zontal plane, without a change in the angular velocity of the 
wheel. This change is the equivalent of the introduction of a 
new angular velocity about an axis perpendicular to the plane 
of the paper. This new angular velocity, compounded with 
the angular velocity about OA, gives rise, as before, to a change 


FiG. 19. 


with OQ; because from the triangles we have The 


36] MECHANICS OF MASSES. 55 


in the direction of the axis without a change in the angular 
velocity of the wheel; and this change in direction is such as 
to oppose the angular acceleration about CD, introduced by 
the weight at B. The system will revolve in a horizontal plane 
about O as a centre. 

Another explanation, leading to the same results, has been 
given by Poggendorff. As has already been stated, it requires 
the application of a force to change the direction a 
of the axis of a rotating body. This force is ex- 
pended in changing the direction of motion of the 
component parts of the body. Poggendorf’s ex- 
planation of the movements of the gyroscope is ¢ 
based on the action of couples formed by these 
separate forces. p 

Let Fig. 20 represent the rotating wheel of the 
former diagram, the axle being supposed to be Be 
nearly horizontal. If the weight be hung at the Nae 
point é¢, it tends to turn the wheel about a horizontal axis CD. 
The particles moving at A and at Ain the plane CD offer no 
resistance to this change. Those at C moving downwards, and © 
those at D moving upwards, act otherwise. The forces ex- 
pressed by their momentum in the directions Cpand Dg are re- 
solved into two each, one of them in the new plane assumed 
by the wheel, and the other at right angles to it. It will be 
seen that the latter component acts at C towards the right, 
and at D towards the left. There is thus set up a couple act- 
ing to turn the system about the axis AZ counter-clockwise, 
as seen from A. As soon as this rotation begins, the particles 
moving at A out of the paper, and at & through the paper, are 
turned out of their original directions, and there arises another 
couple, of which the component at 4 is directed towards the 
left, and at B towards the right. This couple tends to cause 
the system to rotate about the axis CD counter-clockwise, as 
seen from C, and thus to oppose the tendency to rotation due 
to the weight at e. 


56 ELEMENTARY PHYSICS. [36 


All other points on the wheel except those in the lines AB 
and CD, are turned out of their paths by both rotations; and 
therefore components of the forces due to their motions ap- 
pear in both couples in the final summation of effects. The 
result of the existence of these couples is a movement such as 
has already been described. 

36. Moment of Inertia. —The moment of inertia of any 
body about an axis is defined as the summation of the products 
of the masses of the particles making up the body into the 
squares of their respective distances from the axis. 

This product is the measure of the importance of the body’s 
inertia with respect to rotation, and is proportional to the ki- 
netic energy of the body having a given angular velocity about 
the axis; for, if any particle m, at a distance 7 from the axis, 
rotate with an angular velocity w, its velocity is r@ and its ki- 
netic energy is $mw’r*. The whole kinetic energy of the body 
is, therefore, }w*Smr*; and since we have assumed 4a” to be 
given, 2r* is proportional to the kinetic energy of the ro- 
tating body. <A distance & such. that*3h*ow* 2m = 1@*Z2mr 
is called the radius of gyration, and is the distance at 
which a mass equal to that of the whole body must be concen- 
trated to possess the same moment of inertia as the body 


possesses. 
The formula for moment of inertia is 


f= 27: (18) 


and its dimensions are J/L’. 

The moment of inertia of a body with reference to an axis 
passing through its centre of inertia being known, its moment 
of inertia with reference to any other axis, parallel to this, is 
found by adding to the moment of inertia already known, the 
product of the mass of the body into the square of the distance 
of its centre of inertia from the new axis of rotation. For if 


363 MECHANICS OF MASSES. 57 


the centre of inertia of the body of which we know the moment 
of inertia be C, and if 7 be any particle of that body, and if 
O be the new axis to which the moment of inertia is to be re- 
ferred, making the construction as in Fig. 21, we have 


tf 


mak Mls ae? S les al 


Multiplying by the mass m, performing a similar operation 
for every particle of the body, and summing the results, we 
have 


[= 2a’ + Zmr? + 22=mr 6. 


O 
The term 227r 6 on the right van- | 
ishes, for we may write it 27,275; and, Ube 
since C is the centre of inertia, 270 is 
zero ($ 33). Therefore pe 
[= 1+ Mr}. (19) 


This equation embodies the proposition which was to be 
proved. 

The moment of inertia of the simple geometrical solids 
may be found by reckoning the moments of inertia for the 
separate particles of the body, and summing the results. We 
will show how this may be done in a few simple cases. 

(1) To find the moment of inertia of a very thin rod AB, 
of length 2/’ and mass 27’, about an axis +x’, passing through 
the middle point: 

Suppose the half-length to be divided into a very large 


number z of equal parts. The mass of each will be ~ he 


; ae 2 
distance of the first from the axis is oe of the second ooh etc. 


58 ELEMENTARY PHYSICS. [36 
Their moments of inertia are 


yn’ % hfs mn’ . ya mn , te 
qs manag cn Bay > Aa e e e ee U heer =4 
2 nr’ Hn 4p n nm? 


and the moment of inertia of the half-rod is 


I+ 4t9.. +2) 


nu 


jhe 


But (1 +4+9...-+ 2’), where x is indefinitely large, is * 


hence J’ = a 


Fic. 22. FIG. 23. 


If 7 equal the whole length of the rod, 7 the whole mass, 
and / the entire moment of inertia, 


Y Gms as (20) 


(2) To find the moment of inertia of a thin plate AB (Fig. 
23), of length 7 and breadth 20’, about an axis perpendicular 
to it and passing through its centre: 


36] MECHANICS OF MASSES. 59 


Suppose the half-plate to be divided into z rods, parallel to 


its length: each rod will have a length 7 and a breadth . 


/ / 


Their distances from the axis are spires: etc. Let mm be the 


mass of the plate. The moment of inertia of each rod, with 
respect to an axis passing bah its centre of inertia and 


be 
perpendicular to its length, 1S 3 — x - wehbe moments of in- 


ertia of the several rods about ate eae axis 4x’ are 


m [{l? a m (a | 
es Tea, ee) See Ae) Cte, 


72 2 


and the moment of inertia of the half-plate is 


MM 


Mx gt Stato... ta)= (54%) 


20 n 
and of the whole plate equals 


2 B? 


ag (21) 


A parallelopiped of which the axis is x4’ may be supposed 
to be made up of an infinite number of plates, such as AB. 
Its moment of inertia will be the moment of inertia of one 
plate multiplied by the number of plates; or, if JZ is the mass 
of the parallelopiped, its moment of inertia is 


2s ABR A soi 


60 ELEMENTARY PHYSICS. [37 


The moment of inertia of any body, however irregular in 
form or density, may be found experimentally by the aid of 
another body of which the moment of inertia can be computed 
from its dimensions. We will anticipate the law of the pendu- 
lum, which has not been proved, for the sake of clearness. 
The body of which the moment of inertia is desired is set 
oscillating about an axis under the action of a constant force f. 
Its time of oscillation is, then, 


= 7, 


where / is the moment of inertia. 
If, now, another body, of which the moment of inertia can 
be calculated, be joined with the first, the time of oscillation 


alters to 
PA Lia 
Z, =— n/t, 


where J, is the moment of inertia of the body added. Com- 
bining the two equations, we obtain, as the value of the 
moment of inertia desired, 


i Pp ER (23) 


37. Central Forces.—If the velocity or direction of motion 
of a moving body in any way alter, we conceive it to be acted 
on by some force. In certain cases the direction of this force, 


37] MECHANICS OF MASSES. Or 


and the law of its variation with the position of the body, may 
be determined by considering the path or orbit traversed by 
the body and the circumstances of its motion. 

We shall illustrate this by a few propositions, selected on 
account of their applicability in the establishment of the 
theory of universal gravitation. The proofs are substantially 
those given by Newton in the “ Principia.” 

Proposition [.—1i the radius vector, drawn from a fixed point 
to a body moving ina curve, describe equal areas in equal 
times, the force which causes the body to move in the curve 
is directed towards the fixed point. 

Let us suppose the whole time divided into equal periods, 
during any one of which the body is not acted on by the force. 
It will, in the first period, move over a space represented by a 
straight line, as JB (Fig. 24). Inthe second period, it would, 
if unhindered, move over.an equal space 4D and in the same 
line. Let us suppose it, however, deflected by a force acting 
instantaneously at the point &. It will move in a line BC such 
that, by hypothesis, triangle OBA = triangle OBC. Now, 
triangle ODB also = triangle OAS, therefore triangle OCB = 
triangle ODB, and CD is parallel to C4. Complete the paral- 
lelogram CDBE; then it is evident that the motion BC is | 
compounded of the motions BD and SE; and since forces are 
proportional to the motions they 
occasion, the force acting at B is 
proportional to BZ, and is directed 
along the line BO. If now the 
periods into which the whole time peat 
is divided become indefinitely small, in the limit the broken 
line ABC approaches indefinitely near to a curve, and the force 
which causes the motion in the curve is always directed to the 
centre O. 

Proposition I[.—\f a body move uniformly in a circle, the 


62 ELEMENTARY PHYSICS. [37 


force acting upon it varies as its mass and the square of its 
“ velocity directly, and as the radius of the 
d circle inversely. 

If the body m move in a circle (Fig. 25) 
with a constant angular velocity, and pass 
over, in any very small time ¢, the arc ad, 


i which is so small that it may be taken 
equal to its chord, the motion may be 
resolved into two components ad and ae, 

: one tangent and the other normal to the 

Ba: 25: arc. Now jf, the acceleration towards O, 


being constant for that small time, we have 

$= a 
The angle ade is a right angle, and therefore, by similar tri- 
angles, we have 


ad? 


ae 


But ae = 27; ad represents the space traversed in the time Z, 
eee et : 
and in the limit > tepresents 7, the velocity in the circle. 


From the previous reasoning ac represents 4 /#’; whence 


Pe es is 
PIM a 
and 
2 
mv 
mf = ., 
iy r 


Corollary I.—If two bodies revolve about the same centre, 
and the squares of their periodic times be in the same ratio as 
the cubes of the radii of their respective orbits, the forces 


37] MECHANICS OF MASSES. 63 


acting on them will be inversely as the squares of their radii, 
and conversely. For, if 7 and7, represent the periodic times 
of the two bodies moving in circles of radii 7 and 7, with ve- 


locities v and v,, then, by hypothesis, 


LF ROP La 


. — * ee 3 . 6 
/ 
whence 
(NSD ima ile a 
Now 
ve 16 vu ° v; 
. i — r . r, 9 
whence 


ierns Fy 9, 


Corollary I1,—The relation of Corollary I. holds with refer- 
ence to bodies describing similar parts of any similar figures 
having the same centre. In the application of the proof, 
however, we must substitute for uniform velocity the uniform 
description of areas; and instead of radii we must use the 
distances of the bodies from the centre. The proof is as fol- 
lows: 

If Dand D, represent the radii of curvature of the paths of 
the two bodies, R and #, the distances of the bodies from the 
centre of force, then, by hypothesis, letting A represent the 
area described in one period of time, 


from the similarity of figures. 


64 ELEMENTARY PHYSICS. [37 


Now 
hence 


and 


TS, a ae oe 


Proposition ITT—Ilf a body move in an ellipse, the force 
acting upon it, directed to the focus of the ellipse, varies in- 
versely as the square of the radius 
vector. 

Suppose the body moving in 
the ellipse to be at the point P 
(Fig. 26), and the force to act 
upon it along the radius vector 
SP. At the point / draw sthe 
tangent PR, and from a point Q 
on the ellipse draw the line Qu, 

ba 9: cutting SP in +, and complete the 
parallelogram PRQ*. From Q draw QT perpendicular to SP. 
Also draw the diameter GP and its conjugate DX. The force 
which acts on the body, causing it to leave the tangent PA and 
move in the line PQ, acts along SP, and in a time ¢# (supposed 
very small) causes the body to move in the direction SP over 
the space Pr; and since, in the small time considered, it may 
be assumed constant, 


We ham PA 
whence 
Pye af 


37] MECHANICS OF MASSES. 65 


Again: the area described by the radius vector in the time 


he OT, 


: S 
¢ is equal to — . 


; and if A represent the area described 
in unit time, 

Pte winOe: 

Sires aT 


At 


Equating these values of ¢, we obtain 


SPROI PA abr 
4A’ oe ve ? 


whence 


From Proposition I., the value of A is constant for any 


F 
part of the ellipse. We shall now show that ae is also 


Ody 
constant. 
From similar triangles, 


AEE Bed Noemet 0 AIR 2B 
or, since by a property of the ellipse PE = AC, 


Pe Po = AG bho. 


Again, by another property of the ellipse, 


Gan Pes: OU PO SOL. 


66 ELEMENTARY PHYSICS. [37 


_ If, now, we consider the time ¢ to become indefinitely small, 
in the limit, Pand Q approach indefinitely near; whence 


Oy = Oe and pire tei 
The last proportion then becomes 
VL OE ES fy EOL oH vemos Tal tS ONE Bee 
Again, from similar triangles, 
On: Ode PEA 
and from another property of the ellipse, 


ACPI ODAC. 
whence 
OZ OT =e: 


Combining these proportions, 


Pe Pon Cee 


Po OL eee eles 
OO Te = Canes 
we obtain, finally, 


Pe OT pe ACH bee 


; 2 dw 
that , since Adc and CB are constant, or is constant. 


We have now shown that, in the expression for the value 

of the force on the body at any point in the ellipse, all the 
I 

factors are constant except -<=;. The force, therefore, varies 


S: 


inversely as the square of the radius vector. 


CHARTER all: 
MASS ATTRACTION. 


38. Mass Attraction.—The law of mass attraction was the 
first generalization of modern science. In its most complete 
form it may be stated as follows :— 

Between every two material particles in the universe there 
is a stress, of the nature of an attraction, which varies directly 
as the product of the masses of the particles, and inversely as 
the square of the distance between them. This law is some- 
times called the law of uzztversal attraction and sometimes the 
law of gravitation. 

Some of the ancient philosophers hada vague belief in the 
existence'of an attraction between the particles of matter. 
This hypothesis, however, with the knowledge which they 
possessed, could not be proved. The geocentric theory of the 
planetary system, which obtained almost universal acceptance, 
offered none of those simple relations of the planetary motions 
upon which the law was finally established. It was not until 
the heliocentric theory, advocated by Copernicus, strengthened 
by the discoveries of Galileo, and systematized by the labors 
of Kepler, had been fully accepted, that the discovery of the 
law became possible. | 

In particular, the three laws of planetary motion published 
by Kepler in 1609 and 1619 laid the foundation for Newton’s 
demonstrations. The laws are as follows :— 

I. The planets move in ellipses of which one focus is situ- 
ated at the sun. 

II. The radius vector drawn from the sun to the planet, 
sweeps out equal areas in equal times. 


© 


68 ELEMENTARY PHYSICS. [38 


III. The squares of the periodic times of the planets are 
proportional to the cubes of their distances from the sun. 

Kepler could give no physical reason for the existence of 
such laws. Later in the century, after Huyghens had discov- 
ered certain theorems relating to motion in a circle, it was. 
seen that the third law would hold true for bodies moving in 
concentric circles, and attracted to the common centre by 
forces varying inversely as the squares of the radii of the cir- 
cles. Several English philosophers, among them Hooke, 
Wren, and Halley, based a belief in the existence of an attrac- 
tion between the sun and the planets upon this theorem. 

The demonstration was by no means a rigorous one, and 

was not generally accepted. It was left for Newton to show 
that not only the third, but all, of Kepler’s laws were com- 
pletely satisfied by the assumption of the existence of an 
attraction acting between the sun and the planets, and vary- 
ing inversely as the square of the distance. His propositions 
are substantially given in § 37. 
- Newton also showed that the attraction holding the moon 
in its orbit, which is presumably of the same nature as that 
existing between the sun and the planets, is of the same nature 
as that which causes heavy bodies to fall to the earth. This 
he accomplished by showing that the deviation of the moon 
from arectilineal path is suchas should occur if the force which 
at the earth’s surface is the force of gravity were to extend 
outwards to the moon, and vary in intensity inversely as the 
square of the distance. 

Two further steps were necessary before the final generali- . 
zation could be reached. One was,to show the relation of the 
attraction to the masses of the attracting bodies; the other, to 
show that this attraction exists between all particles of matter, 
and not merely, as Huyghens believed, between those particles 
and the centres of the sun and planets. 

The first step was taken by Newton. By means of pen- 


39] MASS ATTRACTION. | 69 


dulums having the same length, but with bobs of different 
materials, he showed that the force acting ona body at the 
earth’s surface is proportional to the mass of the body, since 
all bodies have the same acceleration. He further brought 
forward, as the most satisfactory theory which he could form, 
the general statement that every particle of matter attracts 
and is attracted by every other particle. 

The experiments necessary for a complete verification of this 
last statement were not carried out by Newton. They were 
performed in 1798 by Cavendish. His apparatus consisted 
essentially of a bar furnished at both ends with small leaden 
balls, suspended horizontally by a long fine wire, so that it 
turned freely in the horizontal plane. Two large leaden balls 
were mounted ona bar of the same length, which turned about 
a vertical axis coincident with the axis of rotation of the sus- 
pended bar. The large balls, therefore, could be set and 
clamped at any angular distance desired from the small balls. 
The whole arrangement was enclosed in a room, to prevent all 
disturbance. The motion of the suspended system was ob- 
served from without by means of a telescope. Neglecting as 
unessential the special methods of observation employed, it is 
sufficient to state that an attraction was observed between the 
large and small balls, and was found to be in accordance with 
the law as above stated. 

39. Measurement of the Force of Gravity.—When two 
bodies attract one another, their relative motions are deter- 
mined by Newton’s third law. In the case of the attraction 
between the earth and a body near its surface, if we adopt a 
point on the earth’s surface as the fixed point of reference, 
the acceleration of the body alone need be considered. Since 
the force acting upon it varies with its mass, and since 
its gain in momentum also varies with its mass, it follows 
that its acceleration will be constant, however its mass may 
vary. We may, therefore, obtain a direct measure of the 


70 . ELEMENTARY PHYSICS. 13> 


earth’s attraction, or of the force of gravity, by allowing a body 
to fall freely,and determining its acceleration. It is found 
that a body so falling at latitude 40° will describe in one second 
about 16 08 feet, or 490 centimetres. Its acceleration is there- 
fore 32.16 in feet and seconds or 980 in centimetres and seconds. 
We denote this acceleration by the symbol g. 

The force acting on the body, or the wezght of the body, is 
seen at once to be mg, where mm is the mass of the body. 

On account of the difficulties in the employment of this 
method, various others are used to obtain the value of g indi- 
rectly. For example, we may allow bodies to slide down a 
smooth inclined plane, and observe their motion. The force 
effective in producing motion on the plane is g sin @, where @ 
is the angle of the plane with the horizontal; the space trav- 
ersedin the time ¢ is s = $ef’ sin @ By observing s and #, the 
value of g may be obtained. The motion is so much less rapid 
than that of a freely falling body that tolerably accurate ob- 
servations can be made. Irregularities due to friction upon 
the plane and the resistance of the air, however, greatly vitiate 
any calculations based upon these observations. This method 
was used by Galileo, who was the first to obtain a measure 
of the acceleration due to the earth’s attraction. 

The most exact method for determining the value of g is 
based upon observations of the oscillation of a pendu- 
lum. 

A pendulum may be defined as a heavy mass, or 
bob, suspended from a rigid support, so that it can 
oscillate about its position of equilibrium. 

In the szmple, or mathematical, pendulum, the bob 

9 is assumed to be a material particle, and to be sus- 
pended by a thread without weight. If the bob be 

Fic. 27. stationary and acted on by gravity alone, the line of 
the thread will be the direction of the force. If the bob be 
withdrawn from the position of equilibrium (Fig. 27), it will be 


41] MASS ATTRACTION. 71 


acted on by a force at right angles to the thread, in a direction 
opposite that of the displacement, expressed by — g sin ¢, 
where ¢ is the angle between the perpendicular and the new 
position of the thread. 

The force acting upon the bob at any point in the circle 
of which the thread is radius, if it be released, and allowed 
to swing in that circle, varies as the sine of the angle be- 
tween the perpendicular and the radius drawn to that point. 
If we make the oscillation so small that the arc may be sub- 
stituted for its sine without sensible error, the force acting on 
the bob varies as the displacement of the bob from the point 
of equilibrium. ; 

A body acted on by a force varying as the displacement of 
the body from a fixed point will have a simple harmonic mo- 
tion about its position of equilibrium. 

Hence it follows that the oscillations of the pendulum are 
symmetrical about the position of equilibrium. The bob will 
have an amplitude on the one side of the vertical equal to that 
which it has on the other, and the oscillation, once set up, will 
continue forever unless modified by outside forces. 

The importance of the pendulum as a means of determin- 
ing the value of g consists in this: that, instead of observing 
the space traversed by the bob in one second, we may observe 
the number of oscillations made in any period of time, and de- 
termine the time of one oscillation; from this, and the length 
of the pendulum, we can calculate the value of g. The errors 
in the necessary observations and measurements are very slight 
in comparison with those of any other method. 

40. Formula for Simple Pendulum.—The formula con- 
necting the time of oscillation with the value of gis obtained 
as follows: ‘The acceleration of the bob at any point in the 
arc is, as we have seen, — g sin ¢, or — g@ if the arc be very 
small. The acceleration in a simple harmonic motion is 


2 


47 2nt 


ace 73 a COS ye 


72 ELEMENTARY PHYSICS, [41 


Since the bob has a simple harmonic motion, we may equate 
these expressions: hence 


2 


bx fF 9 cin 2H 
SP =~ Fil COS “Fe. 


at 
But @ cos ae is the displacement of the point having the 


simple harmonic motion, and is therefore equal to Z@, if 7 rep- 
resent the length of the thread: hence 


Anl 
Sa ay pa 


T= am/ 
4 


In this formula 7 represents the time of a double oscillation. 
It is customary to consider as a unit, the time of a single oscil- 
lation, when the formula becomes 


ta ay/ 2 (24) 


41. Physical Pendulum.—Any pendulum fulfilling the re- 
quirements of the foregoing theory is, of course, unattainable 
in practice. We may, however, calculate, from the known di- 
mensions and mass of the portions of matter making up the phy- 
sical pendulum, what would be the length of a simple pendulum 
which would oscillate in the same time. It is clear that there 
must be some point in every physical pendulum the distance 
of which from the point of suspension is equal to the length of 
the corresponding simple pendulum; for the particles near the 
point of suspension tend to oscillate more rapidly than those 


from which 


4] MASS ATTRACTION. ee 


more remote, and the time of oscillation of the system, if it be 
rigid, will be intermediate between the times of oscillation 
which the particles nearest to,and most remote from, the point 
of suspension would have if they were oscillating freely. There 
will, therefore, be some one particle of which the proper rate 
of oscillation is the same as that of the whole pendulum. Its 
distance from the point of suspension is the length sought. 

In determinations of the value of g by observations upon 
the time of oscillation of a pendulum, the length of the equiva- 
lent simple pendulum may be known in either of two ways. 

(1) The pendulum may be constructed in such a manner 
that its moment of inertia and the position of its centre of 
gravity may be calculated. From these data the required 
length is readily obtained. 

When the pendulum oscillates, each of its particles de- 
scribes a simple harmonic motion, and passes through the 
mid-point of its path at the time that the pendulum passes 
through its position of equilibrium. The velocity of each par- 
ticle at the mid-point of its path can therefore be expressed by 
— = where 7 is the period of a complete oscillation, and @ 
is an amplitude differing for each particle considered. Repre- 
senting the distance of any particle from the axis of suspension 
by 7, and the greatest value of the angular displacement of the 
pendulum by ¢, we have a=rg@. Hence the angular velocity 
of each particle, and therefore of the pendulum, is expressed 


b Ae The kinetic energy of a body, rotating about an 


axis with an angular velocity w, has been shown in § 36 to 


2 


@ : ‘ y ; : 
be expressed by amr: Substituting in this expression the 


value obtained for the angular velocity of the pendulum, we 


2 42 


7 . e . 
obtain 33mrXAZe as the expression for the kinetic energy of 


74 ELEMENTARY “PHYSICS. [4r 


the pendulum at the lowest point of its arc. At this point the 
pendulum possesses no potential energy. Its kinetic energy 
at this point must therefore be equal to its potential energy 
at the highest point of its arc, where it posesses no kinetic 
energy. If we represent by J7 the mass of the pendulum, and 
by & the distance of the centre of gravity from the point of 
suspension, A¢@ represents the distance traversed by the centre 
of gravity between the highest and the lowest points of its arc, 
and $//¢¢ represents the average force acting on the centre of 
gravity between those points to produce rotation. The poten- 
tial energy of the pendulum at the highest point of its arc is, 
therefore, $//Rg¢’. Hence we have 


2 A2 


grt =tMRg¢’; 
whence 
amr 
vis ae 2 
L= 274 | aFR- 2 (25) 


This is the time of oscillation of a simple pendulum of which 


ee 
the length is =e Therefore the moment of inertia of any 
physical pendulum divided by the product of its mass into the 
distance of its centre of.gravity from the axis of suspension gives 
the length of the equivalent simple pendulum. An axis paral- 
lel to the axis of suspension, passing through the point on the 
line joining the axis of suspension with the centre of gravi- 
mr 
MR 
pension, is called the axzs of osctllation. 

A pendulum consisting of a heavy spherical bob suspended 
by a cylindrical wire was used by Borda in his determinations 
of the value of g The moment of inertia and the centre of 


ty of the pendulum and distant from the axis of sus- 


41] MASS ATTRACTION. 75 


gravity of the system were easily calculated, and the length of 
the simple pendulum to which the system was equivalent was 
thus obtained. 

(2) We may determine the length of the equivalent simple 
pendulum directly by observation. The method depends upon 
the principle that, if the axis of oscillation be taken as the 
axis of suspension, the time of oscillation will not vary. The 
proof of this principle is as follows: 

Let x and /—~7 represent the distances from the centre 
of gravity to the axis of suspension and of oscillation re- 
spectively, # the mass of the pendulum, and / its moment 
of inertia about its centre of gravity. Then, since the 
moment of inertia about the axis of suspension is /-+- mr’, we 
have } 


yas I+ mr 


mr 


When the pendulum is reversed, we have 


7 T+ m(l—~r)’ 


mi—r) 


From the first equation we have J = mr(/—r), which 
value substituted in the second gives, after reduction, 
J =/; that is, the length of the equivalent simple pen- [] 
dulum, and consequently the time of oscillation when 
the pendulum swings about its axis of suspension, is the 
same as that when it is reversed, and swings about its 
former axis of oscillation. 

A pendulum (Fig. 28) so constructed as to take 
advantage of this principle was used by Kater in his 
determination of the value of 7; and this form is known, 


ad F e 8. 
in consequence, as Kater’s pendulum, pi 


* 


76 ELEMENTARY PHYSICS. [42 


42. The Balance.—The comparison of masses is of such 
frequent occurrence in physical investigations that it is im- 
portant to consider the theory of the balance and the methods 
of using it. 

To be of value the balance must be accurate and sensitive; 
that is, it must be in.the position of equilibrium when the 
scale-pans contain equal masses, and it must move out of that 
position on the addition to the mass in one pan of a very small 
fraction of the original load. These conditions are attained 
by the application of principles which have already been 
developed. 

The balance consists:essentially of a regularly formed beam, 
poised at the middle point of its length upon knife edges 
which rest on agate planes. From each end of the beam is 
hung a scale-pan in which the masses to be compared are 


Fic, 29. 


placed. Let O (Fig. 29) be the point of suspension of the 
beam; A, &, the points of suspension of the scale-pans; C, the 
centre of gravity of the beam, the weight of which is W. 
Represent OA = OB by J, OC by d, and the angle OAB by a. 

If the weight in the scale-pan at A be /, and that in the 
one at B be P-+-f, where # is a small additional weight, the 
beam will turn out of its original horizontal position, and as- 
sume anew one. Let the angle COC, through which it turns, 
be designated by #. Then the moments of force about O are 
equal; that is, 


(P+ p)Z.cos (a+ £) = Pl.cos (a — #) + Wad.sin B; 


42] MASS ATTRACTION. irs 


from which we obtain, by expanding and transposing, 
Re (26) 


The conditions of greatest sensitiveness are readily deduci- 
ble from this equation. So long as cos a is less than unity, it 
is evident that tan f, and therefore #, increases as the weight 
2P of the load diminishes. As the angle a becomes less, the 
value of £ also increases, until, when A, O, and # are in the 


same straight line, it depends only on and is independ- 


pl 

Wa’ 
ent of the load. In this case tan f increases as d, the distance 
from the point of suspension to the centre of gravity of the 
beam, diminishes, and as the weight of the beam W dimin- 
ishes. To secure sensitiveness, therefore, the beam must be 
as long and as light as is consistent with stiffness, the points 
of suspension of the beam and of the scale-pans must be very 
nearly in the same line, and the distance of the centre of 
gravity from the point of suspension of the beam must be as 
small as possible. Great length of beam, and near coincidence 
of the centre of gravity with the axis, are, however, incon- 
sistent with rapidity of action. The purpose for which the 
balance isto be used must determine the extent to which these 
conditions of sensitiveness shall be carried. 

Accuracy is secured by making the arms of the beam of 
equal length, and so that they will perfectly balance, and by 
attaching scale-pans of equal weight at equal distances from 
the centre of the beam. 

In the balances usually employed in physical and chemical 
investigations, various means of adjustment are provided, by 
means of which all the required conditions may be secured. 
The beam is poised on knife edges; and the adjustment of its 
centre of gravity is made by changing the position of a nut 


78 ELEMENTARY PHYSICS. [42 


which moves on a screw, placed vertically, directly above the 
point of suspension. Perfect equality in the moments of force 
due to the two arms of the beam is secured by a similar hori- 
zontal screw and nut placed at one end of the beam. The 
beam is a flat rhombus of brass, large portions of which are 
cut out so as to make it as light as possible. The, knife edge 
on which the beam rests, and those upon which the scale-pans 
hang, are arranged so that, with a medium load, they are all 
nearly in the same line. A long pointer attached to the beam 
moves before a scale, and serves to indicate the deviation of 
the beam from the position of equilibrium. If the balance be 
accurately made and perfectly adjusted, and equal weights 
placed in the scale-pans, the pointer will remain at rest, or will 
oscillate through distances regularly diminishing on each side 
of the zero of the scale. 

If the weight of a body is to be determined, it is placed in 
one scale-pan, and known weights are placed in the other un- 
til the balance is in equilibrium or nearly so. The final deter- 
mination of the exact weight of the body is then made by one 
of three methods: we may continue to add very small weights 
until equilibrium is established; or we may observe the devia- 
tion of the pointer from the zero of the scale, and, by a table 
prepared empirically, determine the excess of one weight over 
the other; or we may place a known weight at such a point 
on a graduated bar attached to the beam that equilibrium is 
established, and find what its value is, in terms of weight 
placed in the scale-pan, by the relation between the length of 
the arm of the beam and the distance of the weight from the 
middle point of the beam. 

If the balance be not accurately constructed, we can, never- 
theless, obtain an accurate value of the weight desired. The 
method employed is known as Borda’s method of double 
weighing. The body to be weighed is placed in one scale-pan, 
and balanced with fine shot or sand placed in the other. It is 


43] MASS ATTRACTION. 79 


then replaced by known weights till equilibrium is again estab- 
lished. It is manifest that the replacing weights represent 
the weight of the body. 

If the error of the balance consist in the unequal length of 
the arms of the beam, the true weight of a body may be ob. 
tained by weighing it first in one scale-pan and then in the 
other. The geometrical mean of the two values is the true 
weight; for let 7, and /, represent the lengths of the two arms of 
the balance, Pthe true weight, and P, and P, the values of the 
weights placed in the pans at the extremities of the arms of 
femethsveand 2, which balance it. Jhen //,= PZ, and ff, = 
P.J,; from which 


D teen, | ahd ope 


43. Density of the Earth.—One of the most interesting 
problems connected with the physical aspect of gravitation is 
the determination of the density of the earth. It has been 
attacked in several ways, each of which is worthy of consider- 
ation. ; 

The first successful determination of the earth’s density 
was based upon experiments made in 1774 by Maskelyne. He 
observed the deflection from the vertical of a plumb-line sus- 
pended near the mountain Schehallien in Scotland. He then 
determined the density of the mountain by the specific gravity 
of specimens of earth and rock from various parts of it, and 
calculated the ratio of the volume of the mountain to that of 
the earth. From these data the mean specific gravity of the 
earth was determined to be about 4.7. 

The next results were obtained from the experiments of 
Cavendish, in 1798, with the torsion balance already described. 
The density, volume, and attraction of the leaden balls being 
known, the density of the earth could easily be obtained. The 
value obtained by Cavendish was about 5.5. 


80 ELEMENTARY PHYSICS. [43 


-_-~— 


Another method, employed by Carlini in 1824, depends 
upon the use of the pendulum. The time of the oscillation of 
a pendulum at the sea-level being known, the pendulum is 
carried to the top of some high mountain, and its time of os- 
cillation again observed. ‘The value of gas deduced from this 
observation will, of course, be less than that obtained by the 
observation at the sea-level. It will not, however, be as much 
less as it would be if the change depended only on the in- 
creased distance from the centre of the earth. The discrep- 
ancy is due to the attraction of the mountain, which can, 
therefore, be calculated, and the calculations completed as in 
Maskelyne’s experiment. ‘The-value obtained by Carlini by 
this method was about 4.8. 

A fourth method, due to Airy, and employed by him in 
1854, consists in observing the time of oscillation of a pendulum 
at the bottom of a deep mine. By § 29, (1), it appears that 
the attraction of a spherical shell of earth the thickness of which 
is the depth of the mine vanishes. The mean density of the 
earth may, therefore, be determined by the discrepancy between 
the values of g at the bottom of the mine and at the surface. 

Still another method, used by Jolly, consists in determining 
by means of a delicate balance the increase in weight of a 
small mass of lead when a large leaden block is brought 
beneath it. Jolly’s results were very consistent and give as 
the earth’s density the value 5.60. 

These methods have yielded results varying from that ob- 
tained by Airy, who stated the mean specific gravity to be 
6.623, to that of Maskelyne, who obtained 4.7. The most 
elaborate experiments, by Cornu and Baille, by the method of 
Cavendish, gave as the value 5.56. This is probably not far 
from the truth. 

When the density of the earth is known, we may calculate 
from it the value of the constant of mass attraction, that is, the 
attraction between two unit masses at unit distance apart. 


£5] MASS ATTRACTION. SI 


l.cpresenting by D the earth’s mean density, by & the earth’s 
mean radius, and by & the constant of attraction, the mass 
of the earth is expressed by 47R*D. Since by § 29, (2), the at- 
traction of a sphere is inversely as the square of the distance 
from its centre, the attraction of the earth on a gram at a point 
on its surface, or the weight of one gram, is expressed by 


3 


_* D 
Se ara =r k= 4nxRDk. xR is twice the length of the earth’s 


ae or 2 X 10’ centimetres. The value of gat latitude 
40° is 980.11, and from the results of Cornu and Baille we may 
set D equal to 5.56. With these data we obtain “& equal to 
0.000000066 dynes. 

44. Projectiles.—When a body is projected in any direc- 
tion near the earth’s surface, it follows, in general, a curved 
path. If the lines of force be considered as radiating from the 
earth’s centre, this path will be, by Proposition III, $37, an 
eilipse, with one focus at the earth’s centre. If the path pursued 
be so small that the lines may be considered parallel, the centre 
of force is conceived of as removed to an infinite distance, and 
the curve becomes a parabola. 

The fact that ordinary projectiles follow a parabolic path 
was first shown by Galileo, as a deduction from the principle 
which he established,—that a constant force produces a uni- 
form acceleration. The proof is as follows: Suppose the 
body to be projected from the point O taken Y 
as origin, in the direction of the axis OY 
(Fig. 30), making any angle ¢ with OX,a 
vertical axis, and to move with a velocity 


v =. Owing to the accelerating effect of reales: 
raeiop 


gravity, it also moves in the vertical direction OX with a 

velocity v =gt. At any time 7 it will have traversed in 

the direction OY a space y=v¢é, and in the direction OX a 

space x =igt*. The co-ordinates of the position of the body 
6 


82 ELEMENTARY PHYSICS. [44 


at any time ¢ are, therefore, y= vt and =e’. The equa- 
2Ux ee 
tion connecting x and y becomes 7” = eran which is the equa- 
ro} 


tion of a parabola referred to the diameter OX and the tan- 
gent OY. When the body is projected horizontally, the vertex 
of the parabola is at the origin of the motion. The body be- 
gins to approach the earth from the start, and reaches it at 
the same time that it would if allowed to fall freely. 

One special case of importance in the consideration of the 
paths of projectiles is that in which the body moves in a circle. 
It is obvious, that, to bring about this result, the body must 
be projected horizontally with such an initial velocity that the 
acceleration due to the earth’s attraction shall be precisely 
equal to the acceleration toward the centre which is necessary ~ 
in order that the body should move in a circle (Proposition 
II, $37). Hence we must have 


bd Sith 
Ve re oC ? 


where m and JV are the masses of the body and the earth re- 
spectively, & is the earth’s radius, and # the constant of attrac- 
tion. Now zy, the velocity of the body, equals 


27zR 


ia ? 
where Z is the time of one complete revolution, and 
M = énxR'D, 


where D is the earth’s mean density. Substituting these val- 


44] MASS ATTRACTION. 83 


ues, we obtain 


gmmR* _ 4mak'D p 
roe aia als b 


from which 


The result shows that the periodic time of any small body 
revolving about a sphere, and infinitely near its surface, is a 
function of the density only, and does not depend on the radius 
of the sphere. 

Upon this principle Maxwell proposed, as an absolute unit 
of time, the time of revolution of a small satellite revolving in- 
finitely near the surface of a globe of pure water at its maxi- 
mum density. 


Ole basse De hbe 


MOLECULAR MECHANICS. 


CONSTITOULTION, OF “MATTER, 


45. General Properties of Matter.—Besides the proper- 
ties already defined in § 3 as characteristic and essential, we 
find that all bodies possess the properties of compressibility 
and divisibility. 

Compressibility.— All bodies change in volume by change of 
pressure and temperature. If a body of a given volume be 
subjected to pressure, it will return to its original volume when 
the pressure is removed, provided the pressure has not been 
too great. This property of assuming its original volume is. 
called elasticity. The property of. changing volume by the 
application of heat is sometimes specially called dzlatabzlity. 

Divistbtlity— Any body of sensible magnitude may, by 
mechanical means, be divided, and each of its parts may again 
be subdivided; and the process may be continued tiil the re- 
sulting particles become so minute that we are no longer able 
to recognize them, even when assisted by the most perfect ap- 
pliances of the microscope. If the body be one that can be 
dissolved, it may be put in solution, and this may be greatly 
diluted; and in some cases the body may be detected by the 
color which it imparts to the diluent, even when constituting 
so small a proportion as one one-hundred-millionth part of the 
solution. 

46. Molecules.—We are not, however, at liberty to con- 
clude that matter is infinitely divisible. The fact, established 


48] MOLECULAR MECHANICS. : 85 


by observation, that bodies are impenetrable, and the one just 
noted, that they are also compressible, as well as other consid- 
erations, to be adduced later, lead to the opposite conclusion. 
To explain the coexistence of these properties, we are com- 
pelled to assume that bodies are composed of extremely small 
portions of matter, indivisible without destroying their identity, 
called molecules, and that these molecules are separated by in- 
terstitial spaces relatively larger, which are occupied by a highly 
elastic medium called the ether. 

These molecules can be divided only by chemical means. 
The resulting subdivisions are called atoms. The atom, how- 
ever, cannot exist in a free state. The molecule is the physi- 
cal unit of matter, while the atom is the chemical unit. 

47. Composition of Bodies.—It has just been said that 
atoms cannot exist in a free state. They are always combined 
with others, either of the same kind, forming simple substances, 
or of dissimilar kinds, forming compound substances. 

There are about sixty-seven substances now known which 
cannot, in the present state of our knowledge, be decomposed, 
or made to yield anything simpler than themselves. We 
therefore call them szmple substances, elements, or, if we desire 
to avoid expressing any theory concerning them, radicals. It 
is not improbable that some of these will yet be divided, 
perhaps all of them. We can call them elements, then, only 
provisionally. 

48. States of Aggregation.—Bodies exist in three states, 
—the solid, the liquid, and the gaseous. In the sold state the 
form and volume of the body are both definite. In the “guzd 
state the volume only is definite. In the gaseous state neither 
form nor volume is definite. , 

Many substances may, under proper conditions, assume 
either of these three states of aggregation; and some sub- 
stances, as, for example, water, may exist in the three states 
under the same general conditions. 


86 ELEMENTARY PHYSICS. [49 


It is proper to add, however, that there is no such sharp 
line of distinction between the three states of matter as our 
definitions imply. Bodies present all gradations of agerega- 
tion between the extreme conditions of solid and gas; and the 
same substance, in passing from one:state to the other, often 
presents all these gradations. 

49. Structure of Solids.—With the exception of organized 
bodies, all solids may be divided into two classes. The bodies 
of one class are characterized by more or less regularity of 
form, which is called crystalline, those of the other class, ex- 
hibiting no such regularity, are called amorphous. For the 
formation of crystals a certain amount of freedom of motion 
of the molecules is necessary. Such freedom of motion is 
found inthe gaseous and liquid states; and when crystallizable 
bodies pass slowly from these to the solid state, crystallization 
usually occurs. It mayalso occurin some solids spontaneously, 
or in consequence of agitation of the molecules by mechanical 
means, such as friction or percussion. Crystallizable bodies 
are called crystallozds. 

Some amorphous bodies cannot, under any circumstances, 
assume the crystalline form. They are called colloids. 

50. Crystal Systems.—Crystals are arranged by mineralo- 
gists in six systems. 

In the first, or /sometric, system, all the forms are referred 
to three equal axes at right angles. The system includes the 
cube, the regular octahedron, and the rhombic dodecahedron. 

In the second, or Dimetric, system, all the forms are referred 
to a system of three rectangular axes, of which only two are 
equal. 

In the third, or Hexagonal, system, the forms are referred 
to four axes, of which three are equal, lie in one plane, and 
cross each other at angles of 60°. The fourth axis is at right 
angles to the plane of the other three, and passes through their 
common intersection. 


52] MOLECULAR MECHANICS. 87 


The fourth, or Orthorhombic, system is characterized by 
three rectangular axes of unequal length. 

In the fifth, or Monoclinic, system, the three axes are un- 
equal. One of them is at right angles to the plane of the 
other two. The angles which these two make with each other, 
as well as the relative lengths of the axes, vary greatly for 
different substances. 

In the sixth, or 77zclinic, system, the three axes are oblique 
to each other, and unequal in length. 

51. Forces determining the Structure of Bodies.—In 
view of what precedes, it is necessary to assume the existence 
of certain forces other than the mass attraction considered in 
§ 38 acting between the molecules of matter. These forces 
seem to act only within very small or insensible distances, and 
vary with the character of the molecule. They are hence 
called molecular forces. In liquids and solids, there must be a 
force of the nature of attraction, holding the molecules to- 
gether, and a force equivalent to repulsion, preventing actual 
contact. The attractive force is called cohestzon when it unites 
molecules of the same kind, and adhesion when it unites mole- 
cules of different kinds. The repulsive force is probably a 
manifestation of that motion of the molecules which constitutes 
heat. In gases this motion is so great as to carry the molecules 
beyond the limit of their mutual molecular attractions: thus 
the apparent repulsion prevails, and the gas only ceases ex- 
panding when this repulsion is balanced by other forces. 

52. Structure of the Molecule.—The facts brought to 
light in the study of crystals compel us to ascribe a structural 
form to the molecule, determining special points of application 
for the molecular forces. From this results the arrangement 
of molecules, which have the requisite freedom of motion, into 
regular crystalline forms. 


88 ELEMENTARY PHYSICS. [53 


FRICTION. 


53. General Statements.—When the surface of one body 
is made to move over the surface of another, a resistance to 
the motion is set up. ‘This resistance is said to be due to /rae- 
tion between the two bodies. It is most marked when the sur- 
faces of two solids move over one another. It exists, however, 
also between the surfaces of a solid and of a liquid ora gas, and 
between the surfaces of contiguous liquids or gases. When the 
parts of a body move among themselves, there is a similar re- 
sistance to the motion, which is ascribed to friction among the 
molecules of the body. This internal friction is called vzscoszty. 

54. Laws of Friction.—Owing to our ignorance of the ar- 
rangement and behavior of molecules, we cannot form a theory 
of friction based upon mechanical principles. The laws which 
have been found are almost entirely experimental, and are only 
approximately true even in the cases in which they apply. 

It was found by Coulomb that, when one solid slides over 
another, the resistance to the motion is proportional to the 
pressure normal to the surfaces of contact, and is independent 
of the area of the surfaces and of the velocity with which the 
moving body slides over the other. It depends upon the na- 
ture of the bodies, and the character of the surfaces of contact. 
The ratio of the force required to keep the moving body in 
uniform motion to the force acting upon it normal to the sur- 
faces of contact is called the coefficzent of friction. } 

It was shown experimentally by Poiseuille that the rate of 
outflow of a liquid from a vessel through a long straight tube 
of very small diameter is proportional directly to the difference 
in pressure in the liquid at the two ends of the tube, to the 
fourth power of the radius of the tube, and inversely to the 
length of the tube. The flow of liquid under such conditions 
can be determined by mathematical analysis, and it is found 


56] MOLECULAR MECHANICS. 89 


that the results obtained by Poiseuille can only occur if the co- 
efficient of friction between the liquid and the wall of the tube 
be very great. In other words, we may think of the liquid par- 
ticles nearest the wall as adhering to it and forming a tube of 
molecules of the same sort as those of the liquid. The outflow 
then depends only upon the coefficient of viscosity of the liquid. 

From consideration based upon the kinetic theory of gases, 
Maxwell predicted that the coefficient of viscosity in a gas 
would be independent of its density. This prediction has 
been verified by experiment through a wide range of densities. 
For very low densities, it has been shown that this law no 
longer holds true. 

55. Theory of Friction.—The friction between solids is due 
largely, if their surfaces be rough, to the interlocking of pro- 
jecting parts. In order to slide the bodies over one another, 
these projections must either be broken off, or the surfaces 
must separate until they are released. There is also a direct . 
interaction of the molecules which lie in the surfaces of con- 
tact. This appears in the friction of smooth solids, and is the 
sole cause of the viscosity of liquids and gases. That this mo- 
lecular action is of importance in producing the friction of 
solids is seen in the facts that the friction of solids of the same 
kind is greater than that of solids of different kinds, and that it 
requires a greater force to start one body sliding over another 
than to maintain it in motion after it is once started. 


CAPILLARITY. 


56. Fundamental Facts.—If we immerse one end of a 
fine. glass tube having a very small, or capillary, bore in 
water, we observe that the water rises in the tube above its 
general level. We also observe that it rises around the outside 
of the tube, so that its surface in the immediate vicinity of the 
tube is curved. If we immerse the same tube in mercury, the 


surface of the mercury within and just outside the tube, instead 


90 ELEMENTARY PHYSICS. [57 


of being elevated, is depressed. If we change the tube for one 
of smaller bore, the water rises higher and the mercury sinks 
lower within it; but the rise or depression outside the tube 
remains the same. If we immerse the same tube in different 
liquids, we find that the heights to which they ascend vary for 
the different liquids. - Hf, instead of changing the diameter, we 
change the thickness of.the wall of the tube, no variation 
occurs in the amount of elevation or depression; and, finally, 
the rise or depression in the tube varies for any one liquid with 
its temperature. 

57. Law of Force assumed.—It is found that a force 
such as is given by the law of mass attraction is not sufficient 
to produce these phenomena. They can, however, be ex- 
plained if we assume an additional attraction between the 
molecules, such as we have already done. The expression, 
then, of the stress between two molecules 7 and mm’, at dis- 
tance 7, becomes ; 


mu 


OMe 


2 + mm' f(r). 

The only law which it is necessary to assign to the function 
of vy in the second term is, that it is very great at insensible 
distances, diminishes rapidly as 7 increases, and vanishes while 
v, though measurable, is still a very small quantity. For adja- - 
cent molecules this molecular attraction is so much greater 
than the mass attraction, that it is customary, in the discussion 


/ 
mine 
of capillary phenomena, to omit the term A from the ex- 


pression for the force. The distance through which this at- 
traction is appreciable is often called the radius of molecular 
action, and is denoted by the symbol e. It is a very small dis- 
tance, but is assumed to be much greater than the distance 
between adjacent molecules. 


59] MOLECULAR MECHANICS. QI 


58. Methods of Development.—The different methods 
which have been employed to deduce, from this assumed 
attraction, results which could be submitted to experimental 
verification, are worthy of notice. They are distinct, though 
compatible with one another. Young was the first to treat the 
subject satisfactorily, though others had given partial and im- 
perfect demonstrations before him. He showed that a liquid 
can be dealt with as if it were covered at the bounding surface 
with a stretched membrane, in which is a constant tension 
tending to contract it. From this basis he proceeded to 
deduce some of the most important of the experimental laws. 
Laplace, proceeding directly from the law of the attraction 
which we have already given, considered the attraction of a 
mass of liquid ona filament of the liquid terminating at the 
surface, and obtained an expression for the pressure within the 
mass at the interior end of the filament. He also was able, 
not only to account for already observed laws, but to predict, 
in at least one instance, a subsequently verified result. Some 
years later, Gauss, dissatisfied with Laplace’s assumption, with- 
out a priori demonstration, of a known experimental fact, 
treated the subject from the basis of the principle of virtual 
velocities, which in this case is the equivalent of that of the 
conservation of energy. He proved, that, if any change be 
made in the form of a liquid mass, the work done or the energy 
recovered is proportional to the change of surface, and hence 
deduced a proof -of the fact which Laplace assumed, and also 
an expression for the pressure within the mass of a liquid 
identical with his. For purposes of elementary treatment the 
earliest method is still the best. We shall accordingly employ 
the idea of surface tension, after having shown that it may be 
obtained from our first hypothesis. 

59. Surface Tension.—Let us consider any liquid bounded 
by a plane surface, of which the line mz (Fig. 31) is the trace, 
and let the line s’z’ be the trace of a parallel plane ata 


Q2 ELEMENTARY PHYSICS. [59 


distance € from the plane of mz. The liquid is then divided 
into two parts by the plane of m’n’,—the general mass of the 
liquid, and a shell of thickness € between the two planes. 
Then, if we imagine a plane passed through any point within the 
general mass, it is clear that the attraction of the molecules on 
opposite sides of that plane will give rise to a pressure normal 
to it, which will be constant for every direction of the plane; 
for the number of molecules now acting on the point is the 
same in all directions. Let, however, the point chosen be P, 
situated within the shell. With P as acentre, and with radius e¢, 
describe a sphere. Now, it is evident that the number of mole- 


Fic. 31. 


cules active in producing pressure upon the plane through P, 
parallel to zn, is less than that of those producing pressure upon 
the plane through Pnormal to mz. The pressure upon the par- 
allel plane varies as we pass from the mass through the shell, 
from the value which it has within the mass, to zero, which it 
has at the plane #z. From this inequality of pressure in the 
two directions, parallel and normal to the surface, there results 
a stress or tension of the nature of a contraction in the surface. 

Provided the radius of curvature of the surface be not very 
small, this tension will be constant for the surface of each 
liquid, or, more properly, for the surface of separation between 
two liquids, or a liquid and a gas. 


60] MOLECULAR MECHANICS. 93 


60. Energy and Surface Tension.—We may here show 
how the energy of the liquid is related to the surface tension. 
It is plain, that, if the molecules, which by their mutual attrac- 
tions give rise to the surface tension, be forced apart by the 
extrusion from the mass into the shell of a sheet of molecules 
along a plane normal to the surface, work will be done as the 
surface is increased. In every system free to move, move- 
ments will occur until the potential energy becomes a min- 
imum: hence every free liquid moves so that its bounding 
surface becomes as small as possible; that is, it assumes a 
spherical form. This is exemplified in falling drops of water 
and in globules of mercury, and can be shown on a large scale 
by a method soon to be described. If we call the potential 
energy lost by a diminution in the surface of one unit, the 
surface energy per unit surface, we can show that it is numeri- 
cally equal to the surface tension across one unit of length. 

Suppose a thin film of liquid to be stretched on a frame 


B 


A 
Fic. 32. 


ABCD (Fig. 32), of which the part BCD is solid and fixed, and 
the part A is a light rod, free to slide along Cand D. This 
film tends, as we have said, to diminish its free surface. As it 
contracts, it draws A towards &. If the length of A be a,and 
A be drawn towards & over 6 units, then if & represent the 
surface energy per unit of surface, the energy lost, or the work 
done, is expressed by Zad. If we consider the tension acting 


O04 ; LEME LALLY LIP SLOSS [6r 


normal to A, the value of which is 7 for every unit of length, we 
have again for the work done during the movement of A, Zaé. 
From these expressions we obtain at once & = 7; that is, the 
numerical value of the surface energy per unit of surface is 
equal to that of the tension in the surface, normal to any line 
in it, per unit of length of that line. 
61. Equation of Capillarity.—The surface tension intro- 
duces modifications in the pressure within the liquid mass 
($85 seg.) depending upon the curvature of 
agi the surface. Consider any infinitesimal rect- 
/ angle (Fig. 33) on the surface) sepuene 
/ length of its sides be represented by sands, 
| / respectively, and the radii of curvature of 
?/ those sides by Rand &,. Also let d and @, 
represent the angles in circular measure sub- 
tended by the sides from their respective 
\1/ centres of curvature. Now, a tension 7 for 
\ every unit of length acts normal to s and 
Fic. 33. tangent to the surface. The total tension 
across s is then Zs; and if this tension be resolved parallel and 
normal to the normal at the point P, the centre of the rect- 


Ps 
2 


angle, we obtain for the parallel component 7s sin a or, 


. ; Ss 3 
since @, is a very small angle, (ee or Ts. The opposite 
side gives a similar component ; the side sand the side. oppo- 


site it give each a component Zs —; = The total force along 


the normal at P is then 
Tss e arr a 


and since ss, is the area of the infinitesimal rectangle, the force 


62] MOLECULAR MECHANICS. 95 


or pressure normal to the surface at Preferred to unit of sur- 
face is 


I Py 
Tle + zi 


From a theorem given by Euler we know that the sum 


I Fae Wes 
RP + R is constant at any point for any position of the rect- 


angular normal plane sections; hence the expression we have 
obtained fully represents the pressure at P. 

If the surface be convex, the radii of curvature are positive, 
and the pressure is directed towards the liquid; if concave, 
they are negative, and the pressure is directed outwards. This 
pressure is to be added to the constant molecular pressure 
which we have already seen exists everywhere in the mass. If 
we denote this constant molecular pressure by A, the ex- 
pression for the total pressure within the mass is 


K+7(e +R) 


where the convention with regard to the signs of R and Rk 
must be understood. Fora plane surface, the radii of curva- 
ture are infinite, and the pressure under such a surface reduces 
to K. 

62. Angles of Contact.—Many of) the capillary phenom- 
ena appear when different liquids, or liquids and sclids, are 
brought in contact with one another. It becomes, therefore, 
necessary to know the relations of the surface tensions and the 
angles of contact. They are determined by the following 
considerations: 

Consider first the case when three liquids meet along a line. 


96 ELEMENTARY PHYSICS. [62 


Let O represent the point where this line cuts a plane drawn 
at right angles to it. Then the ten- 
sion 7,; of the surface of separation 
of the liquid @ from the liquid 34, 
acting normal to this line, is coun- 
terbalanced by the tensions 7,. and 
T;- of the surfaces of separation of 
aandc,dandc. These tensions are 

Fic. 34. always the same for the three liquids 
under similar conditions of temperature and purity. Knowing 
the value of the tensions, the angles which they make with 
one another are determined at once by the parallelogram of 
forces; and these angles are always constant. 

Similar relations arise if one of the liquids be replaced by a 
gas. Indeed, some experiments by Bosscha indicate that 
capillary phenomena occur at surfaces of separation between 
gases. We need, therefore, in the subsequent discussions, 
make no distinction between gases and liquids, and may use 
the general term fluids. 

If 7,, be greater than the sum of 7), and 7;,,, the angle be- 
tween 7,.and 7; becomes zero, and the 
fluid ¢ spreads itself out in a thin sheet j 
between a and J. Thus, if a drop of oil 
be placed on water, the tension of the j 
surface of separation between the air and 
water is greater than the sum of the ten- / a 
sions of the surfaces between the air and i] 
oil, and between the oiLand water; hence 7 
the drop of oil spreads out over the water y 
until it becomes almost indefinitely thin. 

In the case of two fluids in contact with ]} 
a plane solid (Fig. 35), it is evident that 
when the system is in equilibrium, the 
surface of separation between the fluids @ and 6, making the 
angle 6 with the solid C, is 


FIG. 35. 


63] MOLECULAR MECHANICS. 97 


Sine = Pige + TAgCos 0. 
The angle of contact is then determined by the equation 


6 pull Lac yi ‘Gy 
LOS a By eae ° 


If 7,, be greater than 7,,,-+ 7;,., the équation gives an im- 
possible value for cos & In this case the angle becomes 
evanescent, the fluid 4 spreads itself out, and wets the whole 
surface of the solid. In other cases the value of @ is finite 
and constant for the same substances. Thus, a drop of water 
placed on a horizontal glass plate will spread itself over the 
whole plate; while a small quantity of mercury placed on the 
same plate will gather together into a drop, the edges of which 
make a constant angle with the surface. 

63. Plateau’s Experiments.—The preceding principles 
will enable us to explain a few of the most important experi- 
mental facts of capillarity. 

A series of interesting results was obtained by Plateau from 
the examination of the behavior of a mass of liquid removed 
from the action of gravity. His method of procedure was to 
place a mass of oil in a mixture of alcohol and water, carefully 
mixed so as to have the same specific gravity as the oil. The 
oil then had no tendency to move as a mass, ard was free to 
arrange itself entirely under the action of the molecular forces. 
Referring to the equation of Laplace, already obtained, it 
is evident that equilibrium can exist only when the sum 

I Alt: ; Mia 
etR) is constant for every point on the surface. This is 
manifestly a property of the sphere, and is true of no other 
finite surface. Plateau found, accordingly, that the freely 
floating mass at once assumed a spherical form. This result 
we had previously reached by another method. If a solid 

7 


98 ELEMENTARY PHYSICS. [63 


body—for instance, a wire frame—be introduced into the mass 
of oil, of such a size as to reach the surface, the oil clings to it, 
and there is a break in the continuity of the surface at the 
points of contact. Each of the portions of the surface divided 
from the others by the solid then takes a form which fulfils 


I 
the condition already laid down, that (5 +- z) equals a con- 


stant. Plateau immersed a.wire ring in the mass of oil. So 
long as the ring nowhere reached the surface, the mass re- 
mained spherical. On withdrawing a portion of the oil witha 
syringe, that which was left took the form of two equal 
calottes, or sections of spheres, forming a double convex lens. 
A mass of oil, filling a short, wide tube, projected from it at 
either end in a similar section of a sphere. As the oil was 
removed, the two end surfaces became less curved, then plane, 
and finally concave. 

Plateau also obtained portions of other figures which fulfil 
the required condition. For example, a mass of oil was made 
to surround two rings placed at a short distance from one 
another. Portions of the oil were then gradually withdrawn, 
when two spherical ca/ottes formed, one at each ring, and the 
mass between the rings became a right cylinder. It is evident 
that the cylinder fulfils the required condition for every point 
on its surface. . 

Plateau also studied the behavior of films. He devised a 
mixture of soap and glycerine, which formed very tough and 
durable films; and he experimented with them in air. Such 

‘films are so light that the action of gravity on them may be 
neglected in comparison with that of the surface tension. If 
the parts of the frame upon which the film is stretched be all 
in one plane, the film will manifestly lie in that plane. When, 
however, the frame is constructed so that its parts mark the 
edges of any geometrical volume, the films which are taken up 
by it often meet. Any three films thus meeting so arrange 


64} MOLECULAR MECHANICS. 99 


themselves as to make angles of 120° with one another. This 
follows as a consequence of the proposition which has already 
been given to determine the equilibrium of surfaces of separa- 
tion meeting along a line. If four or more films meet, they 
always meet at a point. 

Plateau also measured the pressure of air in a soap-bubble, 
and found that it differed from the external pressure by an 
amount which varied inversely as the radius of the bubble. 
This follows at once from Laplace’s equation. This measure- 
ment also gives us a means of determining the surface tension ; 
for, from Laplace’s equation, the pressure inwards, due to the 
outer surface, is TS, and the pressure in the same direction 

‘ ; 2 ‘ : 
due to the inner surface is also Th for the film is so thin 
that we may neglect the difference in the radii of curvature of 
the two surfaces: hence the total pressure inwards is and 
if this be measured by a manometer, we can obtain the value 
of T. 

64. Liquids influenced by Gravity.—Passing now to con- 
sider liquid masses acted on by gravity, we shall treat only a 
few of the most important cases, 

If a glass tube having a 
marrow bore be immersed 
perpendicularly in water, the 
water rises in the tube toa 
height inversely proportional 9 
to the diameter of the tube. 

This law is known as /urzn’s 
daw. | 

Let Fig. 36 represent the 
section of a tube of radius 7 Bitsy ace 
immersed in a liquid, the surface of which makes an angle 6 


100 ELEMENTARY PHYSICS. [64 


with the wall. Then if 7 be the surface tension of the liquid, 
the tension acting upward is the component of this surface 
tension parallel to the wall, exerted all around the circumfer- 
ence of the tube. This is expressed by 


anr 7, cos o. 
This force, for each unit area of the tube, is 


2mrI cos 6 

aaa vt ae 

The downward force, at the level of the free surface, making 
equilibrium with this, is due to the weight of the liquid column 
($86). If we neglect the weight of the meniscus, this force 
per unit area, or the pressure, is expressed by dg, where & is 
the height of the column and d@ the density of the liquid. We 
have, accordingly, since the column is in equilibrium, 


227 
apt 6s fice hdg ; 


whence 
oe 2T cos ] 


hatantar i) Se 


and the height is inversely as the radius of the tube. 
If the liquid rise between two parallel plates of length 7, 
separated bya distance 7, the upward force per unit area is 
spend 
siven by the expression Zt 698 6, and the downward pressure 


by hag; whence 


65] MOLECULAR MECHANICS. | 101 


and the height to which the liquid will rise between two such 
plates is equal to that to which it will rise in a tube the radius 
of which is equal to the distance between the plates. 

If the two plates are inclined to one another so as to touch 
along one vertical edge, the elevated surface takes the form of 
a rectangular hyperbola; for let the line of contact of the 
plates be taken as the axis of ordinates, and a line drawn 
in the plane of the free surface of the liquid as the axis of 
abscissas, the elevation corresponding to each abscissa is in- 
versely as the distance between the plates at that point, and 
the elevations are therefore inversely as the abscissas: hence 
the product of any abscissa by its corresponding ordinate is a 
constant. The extremities of the ordinates then mark outa 
rectangular hyperbola referred to its asymptotes. 

65. Liquid Drops in Capillary Tubes.—When a drop of 
liquid is placed in a conical tube, it moves, if the surfaces are 
concave, towards the smaller 
end; if convex, towards the 
larger end. The explanation 
of these movements follows 
readily from the foregoing 
results. In case the surfaces het 
are concave, letting 6 (Fig. 37) be the angle of contact and @ 
the angle of inclination of the wall of the tube to the axis, 7 
and 7, the radii of the tube at the extremities of the drop, 7 
being the smaller of the two, then the expressions for the com- 
ponents of the tensions parallel with the axis acting in both 
cases outwards, are respectively 


OL (6 — a), 
x 
and 
asta oa (6+ a). 


TUF 


102 ELEMENTARY “PITYSICS: [66 


Of these two expressions the former is manifestly greater than 
the latter: hence the tendency of the drop is to move towards 
the smaller end of the tube. 

If we assume that the concave surfaces are portions of 
spheres, of which R& and &#, are the respective radii of curva- 
ture, it follows that y = Rcos(@— a), and7, = R,cos (#4 a); 
hence the expressions for the tensions become a and at 
These are the values of the tensions as determined by Laplace's 
equation, and the movements of the drop might have been in- 
ferred directly from this equation by making the same assump- 
tion. 

If a drop of water be introduced into a cylindrical capillary 
tube of glass, and if the air on the two ends of the drop have 
unequal pressures, the concavities thereby become unequal, 
the one on the side of the greater pressure presenting the 
greater concavity. The drop so circumstanced offers a resist- 
ance to this pressure; and it may, if the pressure be not too 
great, entirely counterbalance it. It is also evident, that, if 
several such drops be introduced successively, with intervening 
air-spaces, the pressure which they can unitedly sustain is equal 
to that which one can sustain multiplied by their number. 
Jamin found that, with a tube containing a large number of 
drops, a pressure of three atmospheres was maintained without 
diminution for fifteen days. 

66. Movements of Solids.—In certain cases the action of 
the capillary forces produces movements in solid bodies partially 
immersed in a liquid. For example, if two plates, which are 
both either wetted or not wetted by the liquid, be partially 
immersed vertically, and brought so near together that the rise 
or depression of the liquid due to the capillary action begins, 
then the plates will move towards one another. In either case 
this movement is explained by the inequality of pressure on 
the two sides of each plate. When the liquid rises between 


68] MOLECULAR MECHANICS. 103 


the plates, the pressure is zero at that point in the column 
which lies in the same plane as the free external surface. At 
every internal point above this the molecules of the liquid are 
in a state of negative pressure or tension, and the plates are 
consequently drawn together. When the liquid is depressed 
between the plates, they are pressed together by the external 
liquid above the plane in which the top of the column between 
the plates lies. When one of the plates is wetted by the liquid 
and the other not, the plates move apart. This is explained 
by noting, that, if the plates be brought near together, the 
convex surface at the one will meet the concave surface at the 
other, and there will be a consequent diminution in both the 
elevation and the depression at the inner surfaces of the plates. 
The elevation and depression at the outer surfaces remaining 
unchanged, there will result a pull outwards on the wetted plate 
and a pressure outwards on the plate which is not wetted; and 
they will consequently move apart. Laplace showed, however, 
as the result of an extended discussion, that, though seeming 
repulsion exists between two plates such as we have just con- 
sidered, yet, if the distance between the plates be diminished 
beyond a certain value, this repulsion changes to an attraction. 
This prediction has been completely verified by the most care- 
ful experiments. 

67. Porous Bodies.—/orous bodies may be considered as 
assemblages of more or less irregular capillary tubes. Thus the 
explanation of many natural phenomena—as the wetting of a 
sponge, the rise of the oil in the wick of a lamp—follows 
directly from the preceding discussion. 


DIFFUSION. 


68. Solution and Absorption.—Many solid bodies, im- 
mersed in a liquid, after a while disappear as solids, and are 
taken up by the liquid. This process is called solutzon. The 


aa’ 


104 FLEMENTARY £AYSICS. [69 


quantity of any body which a unit quantity of a given liquid 
will dissolve at a given temperature, is called its soludzlety in 
that liquid at that temperature. The solubility of a given 
solid varies greatly for different liquids, in many cases being so 
small as to be inappreciable. 

Gases are also taken into solution by liquids. The process 
is usually called absorption. The quantity of gas dissolved in 
any liquid depends upon the temperature, and varies directly 
with the pressure. The solubility of any gas at a given 
temperature and at standard pressure is called its coeficzent of 
absorption at that temperature. 

Gases, in general, adhere strongly to the surfaces of solids 
with which they are in contact. This adhesion is so great, that 
the gases are sometimes condensed so as to form a dense layer 
which probably penetrates to some depth below the surface of 
the solid. ‘The process is called the absorption of gases by 
solids. When the solid is porous, its exposed surface is greatly 
extended, and hence much larger quantities of gas are condensed 
on it than would otherwise be the case. When this condensa- 
tion occurs there is in general a rise of temperature which may 
be so great as to raise the solid to incandescence. Thus, for 
example, spongy platinum, placed in a mixture of oxygen and 
hydrogen, becomes so heated as to inflame it. 

69. Free Diffusion of Liquids.—When two liquids which 
are miscible are so brought together in a common vessel that 
the heavier is at the bottom and the lighter rests upon it ina 
well-defined layer, it is found that after a time, even though no 
agitation occur, they become uniformly mixed. Molecules of 
the heavier liquid make their way upwards through the lighter ; 
while those of the lighter make their way downwards through 
the heavier, in apparent opposition to gravitation. Dzffuszon 
is the name which is employed to designate this phenomenon 
and others of a similar nature. 

When one of the liquids is colored,—as, for example, 


7G). MOLECULAR MECHANICS. 105 


solution of cupric sulphate,—while the other is colorless, the 
progress of the experiment may easily be watched and noted. 
When both liquids are colorless, small glass spheres, adjusted 
and sealed so as to have different but determinate specific 
gravities between those of the liquids employed, may be placed 
in the vessel used in the experiment, and will show by their 
positions the degree of diffusion which has occurred at any 
given time. 

70. Coefficient of Diffusion.—-Experiment shows that the 
amount of a salt in solution which at a given temperature 
passes, in unit time, through unit area of a horizontal surface, 
depends upon the nature of the salt and the rate of change of 
concentration at that surface,—that is, the quantity of a sait 
that passes a given horizontal plane in unit time is «CA, where 
A is the area, C the rate of change of concentration, and «a 
coefficient that depends upon the nature of the substance. 
By rate of change of concentration is meant the difference in the 
quantities of salt in solution measured in grams per cubic 
centimetre, at two horizontal planes one centimetre apart, 
supposing the concentration to diminish uniformly from one 
to the other. It is plain, that, if C and d in the above expres- 
sion be each equal to unity, the quantity of salt passing in 
unit time is x. The quantity «, called the coefficient of diffu- 
sion, is, therefore, the quantity of salt that passes in unit time 
through unit area of a horizontal plane when the difference of 
concentration is unity. Coefficients of diffusion increase with 
the temperature, and are found not to be entirely independent 
of the degree of concentration. 

As implied above, the units of mass and length employed 
in these measurements are respectively the gram and the centi- 
metre; but, since in most cases the quantity of salt that diffuses 
in one second is extremely small, it is usual to employ the day 
as the unit time. 


106 ELEMENTARY PHYSICS. [7x 


71. Diffusion through Porous Bodies.—It was found by 
Graham that diffusion takes place through porous solids, such 
as unglazed earthenware or plaster, almost as though the 
liquids were in direct contact, and that a very considerable 
difference of pressure can be established between the two faces 
of the porous body while the rate of diffusion remains nearly 
constant. 

72. Diffusion through Membranes.—lIf the membrane 
through which diffusion occurs be of a type represented by ani- 
mal or vegetable tissue, the resulting phenomena, though in 
some respects similar, are subject to quite different laws. Col- 
loid substances pass through the membrane very slowly, while 
crystalloid substances pass more freely. It is to be noted 
that the membrane is not a mere passive medium, as is the 
case with the porous substances already considered, but takes 
an active part in the process; and consequently one of the 
liquids frequently passes into the other more rapidly than would 
be the case if the surfaces of the liquids were directly in con- 
tact. 

An explanation of these facts follows if we suppose that 
diffusion of a liquid through a continuous membrane can occur 
only when the liquid is capable of temporarily uniting with the 
membrane, and forming a part of it. Diffusion would then oc- 
-cur by the union of the liquid with the membrane on one face, 
and the setting free of an equal portion on the other. 

If the membrane separate two crystalloids, it often happens 
that both substances pass through, but at different rates. In 
accordance with the usage of Dutrochet, we may say there is 
endosmose of the liquid, which passes more rapidly to the other 
liquid, and exosmose of the latter to the former. The whole 
process is frequently called osmoszs. If the membrane be 
stretched over the end of a tube, into which the more rapid 
current sets, and the tube be placed in a vertical position, the 
liquid will rise in the tube until a very considerable pressure is 


74] MOLECULAR MECHANICS. 107 


attained. Dutrochet called such an instrument an exdosiio- 
meter. 

Graham made use of a similar instrument, which he called 
an osmometer, by means of which he studied, not only the ac- 
tion of porous substances, such as are mentioned above, but 
also that of various organic tissues; and he was able to reach 
quantitative results of great value. Pfeffer has more recently 
made an extended study of the phenomena of osmosis, espec- 
ially in those aspects relating to physiological phenomena. He 
has shown that colloid membranes produced by purely chemi- 
cal means are even more efficient than the organic membranes 
employed by Graham. : 

73. Dialysis.—Upon the principles just set forth Graham 
has founded a method of separating crystalloids from any col- 
loid matters in which they may be contained, which is often of 
great importance in chemical investigations. The apparatus 
employed by Graham consists of a hoop, over one side of which 
parchment paper is stretched so as to constitute a shallow 
basin. In this basin is placed the mixture under investigation, 
and the basin is then floated upon pure water contained in an 
outer vessel. If crystalloids be present, they will in due time 
pass through the membrane into the water, leaving the colloids 
behind. The process is often employed in toxicology for sep- 
arating poisons from ingesta or other matters suspected of 
containing them. It is called dzalyszs, and the substances that 
pass through are said to dzalyse. 

74. Laws of Diffusion of Gases.—Gases obey the same 
elementary laws of diffusion as liquids. The rate of diffusion 
varies inversely as the pressure, directly as the square of the 
absolute temperature, and inversely as the square root of the 
density of the gas. A gas diffuses through porous solids ac- 
cording to the same laws. An apparatus by which this may 
be conveniently illustrated consists of a porous cell, the open 
end of which is closed by a stopper, through which passes a 


108 ELEMENTARY PHYSICS. [75 


long tube. This is placed in a vertical position, with the open 
end of the tube ina vessel of water. If, now, a bell-jar con- 
taining hydrogen be placed over the porous cell, hydrogen 
passes into the cell more rapidly than the air escapes from it: 
the pressure inside is increased, as is shown by the escape of 
bubbles from the end of the tube. If, now, the jar’be removed, 
diffusion outward occurs more rapidly than diffusion inward: 
the pressure within soon becomes less than the atmospheric 
pressure, as is shown by the rise of the water in the tube. 


LLASLICI LY: 


75. Stress and Strain.—When a body is made the medium 


for the transmission of force, the application of Newton’s third _ 


law shows that there is a stress in the medium. This stress is 
always accompanied by a corresponding change of form of the 
body, called a strazn. 

In some bodies equal stresses applied in any: direction pro- 
duce equal and similar strains. Such bodies are zsotropic. In 
others the strain alters with the direction of the stress. These 
bodies are eolotropic. 

According to the molecular theory of matter, the form of a 
body is permanent so long as the resultant of the stresses act- 
ing on it from without, with the interior forces existing be- 
tween the individual molecules of the body, reduces to zero. 
The molecular forces and motions are such that there is a cer- 
tain form of the body for every external stress in which its 
molecules are in equilibrium. Any change of the stress in the 
body is accompanied by a readjustment of the molecules, 
which is continued until equilibrium is again established. 

If the stress tend only to increase or diminish the distance 
between the molecules, it is called a ¢enston or a pressure re- 
spectively; if it tend to slide one line or sheet of molecules 
past another tangentially, it is called a shear or a shearing-stress. 


76] MOLECULAR MECHANICS. 109 


All stresses can be resolved into these two forms. The cor- 
responding changes of shape are called dlatations, compressions, 
and shearing-strains. 

The term pressure is used with several different meanings. 
In order to most clearly present these, we will consider a right 
cylinder, transmitting a stress in the direction of its axis. The 
stress itself is often called the total pressure upon the cylinder. 

If we conceive the cylinder to consist of a great number of 
elementary cylinders of small cross-section, and if the total 
pressure upon any one of them, as here defined, be to the total 
pressure on the whole cylinder as the cross-section of the ele- 
mentary cylinder is to the cross-section of the whole cylinder, 
then it is said'that the pressure on the cross-section is uniform, 
and the pressure on an area in that cross-section is defined as. 
the product of the total pressure on the cylinder into the 
ratio of that area to the cross-section of the cylinder. Further, 
the pressure at a point, in a direction normal to the cross-sec- 
tion, is defined as the ratio of the pressure on an area, taken in 
the cross-section with its centre of inertia at the point, to that 
area, when the area is diminished indefinitely. This definition 
may at once be generalized. The pressure in any given direc- 
tion at apoint ina medium transmitting stress in any manner 
whatever, is the ratio of the pressure on any area, taken nor- 
mal to the given direction and with its centre of inertia at the 
point, to that area, when the area is diminished indefinitely. 

In case a stress exists between two bodies, which acts nor- 
mally across a common surface of contact, the term pressure is 
also used to denote this stress, and the pressures on an area 
and at a point in the surface of contact, are defined exactly as 
above. 

76. Modulus of Elasticity.—If, for a given amount of stress 
between certain limits, a body be deformed by a definite 
amount, which is constant so long as the stress remains con- 
stant, and if, when the stress is removed, the body regain its 


110 ELEMENTARY PHYSICS. (77 


original condition, it is said to be perfectly clastic. Any body 
only partially fulfilling these conditions is said to be imperfectly 
elastic. 

The definition of elasticity in its physical sense, as a prop- 
erty of bodies, has been already given. It is measured by the 
rate of change, in a unit of the body, of the stress with respect 
to the strain. Thus for example, the voluminal elasticity of a 
fluid is measured by the limit of the ratio of any small change 
of pressure to the corresponding change of unit volume. The 
tractional elasticity of a wire under tension is measured by the 
limit of the ratio of any small change in the stretching-weight 
to the corresponding change in unit length. This ratio is called 
the modulus of elasticity, or simply the elasticity of a body, 
and its reciprocal the coeffictent of elasticity. 

77. Modulus of Voluminal Elasticity of Gases.—Within 
certain limits of temperature and pressure the volume of any 
gas, at constant temperature, is inversely as the pressure upon 
it. This law was discovered by Boyle in 1662, and was after- 
wards fully proved by Mariotte. It is known, from its discov- 
erer, as Loyle’s law. 

Thus, if g and g, represent different pressures, v and v, the 
corresponding volumes of any gas at constant temperature, 
then 


Lip, = 
whence 


pv = pV; (27) 


Now, g,v, is a constant which may be determined by choosing 
any pressure g, and the corresponding volume vy, as standards: 
hence we may say, that, at any given temperature, the product 
pv is a constant. The limitations to this law will be noticed 
later. 


77) MOLECULAR MECHANICS. III 


If we draw the curve marked out bya point having its 
ordinate and abscissa so related that 4#y equals a constant, we 
obtain a rectangular hyperbola referred to its asymptotes. Let 
# represent the volume and y the pressure of a quantity of gas. 
Then this curve shows the relation of pressure and volume in 
all their combinations. 

Draw the lines as in Fig. 38, lettings A.C eefy, Ge bieeert 
volumes differing only by a small amount. 

We must first show that AZ is numeri- 
cally equal to the modulus of elasticity. The 


CG 
ratio WG is the voluminal compression per 
unit volume for the increment of pressure 
MmeneGLy 
ee enence, by definition, =< CG is the modulus 


AC 
of elasticity. But, from similarity of tri- 
ees: CD AC: CG. 


Hence we have 


s i a4 
77 oR the modulus cf elasticity. 


Now, since, by construction, the rectangles AZ and /K are 
equal, and the rectangle AX is common to them, the rectangles 
JG and CK are equal, and 


ORIOLES Oph en S148 
By similar triangles, 
EG 7 DGS CARPAL 


whence 
GA? GK = GA: ABZ. 


112 ELEMENTARY PHYSICS. [78 


Now, if the increment of pressure be made indefinitely 
small, so that in the limit D and C coincide, the line CE be- 
comes a tangent to the curve, and GA, GK, are respectively 
equal to CA, CB. CB therefore equals AZ from the last pro- 
portion: hence, in the case of a gas obeying Boyle's law, the 
modulus of elasticity is numerically equal to the pressure. 

The discussion of the experimental facts in connection with 
the elasticity of gases, and the explanation of the apparatus 
founded upon it, will be resumed in a future chapter. 

78. Modulus of Voluminal Elasticity of Liquids.—When 
liquids are subjected to voluminal compression, it is found that 
their modulus of elasticity is much greater than that of gases. 
For at least a limited range of pressures the modulus of 
elasticity of any one liquid is constant, the change in volume 
being proportional to the change in the pressure. The modulus 
differs for different liquids. 

The instrument used to determine the modulus of elasticity 
of liquids is called a prezometer. The first form in which the 
instrument was devised by Oersted, while not the best for ac- 
curate determinations, may yet serve as a type. 

The liquid to be compressed is contained in a thin glass 
flask, the neck of which is a tube with a capillary bore. The 
flask is immersed in water contained in a strong glass vessel 
fitted with a water-tight metal cap, through which moves a 
piston. By the piston, pressure may be applied to the water, 
and through it to the flask and to the liquid contained in it. 

The end of the neck of the small flask is inserted down- 
wards under the surface of a quantity of mercury which lies at 
the bottom of the stout vessel. The pressure is registered by 
means of a compressed-air manometer (§ 96) also inserted in 
the vessel. When the apparatus is arranged, and the piston 
depressed, a rise of the mercuryin the neck of the flask occurs, 
which indicates that the water has been compressed. 

An error may arise in the use of this form of apparatus from 


80] MOLECULAR MECHANICS. Il3 


the change in the capacity of the flask, due to the pressure. 
Oersted assumed, since the pressure on the interior and exterior 
walls was the same, that no change would occur. Poisson, how- 
ever, showed that such a change would occur, and gavea formula 
by which it might be calculated. By introducing the proper 
corrections, Oersted’s piezometer may be used with success. 

A different form of the instrument, employed by Regnault, 
is, however, to be preferred. In it, by an arrangement of stop- 
cocks, it is possible to apply the pressure upon either the 
interior or exterior wall of the flask separately, or upon both 
together, and in this way to experimentally determine the cor- 
rection to be applied for the change inthe capacity of the flask. 

It is to be noted that the modulus of elasticity for liquids 
is so great, that, within the ordinary range of pressures, they 
may be regarded as incompressible. Thus, for example, the 
alteration of volume for sea-water by the addition of the pres- 
sure of one atmosphere is 0.000044. The change in volume, 
then, at a depth in the ocean of one kilometre, where the pres- 
sure is about 99.3 atmospheres, is 0.00437, or about g4, of the 
whole volume. 

79. Modulus of Voluminal Elasticity of Solids.—The 
modulus of voluminal elasticity of solids is believed to be gen- 
erally greater than that of liquids, though no reliable experi- 
mental results have yet been obtained. 

The modulus, as with liquids, differs for different bodies. 

80. Shears.—A strain in which parallel planes or sheets 
of molecules are moved tangentially 
over one another, each plane being 
displaced by an amount proportional 
toits distance from one of the planes 
assumed as fixed, is called a shear. 

To illustrate this definition, let us 
consider a _ parallelopiped, of which Fic. 39. 


the cross-section made at right angles to its sides is a rhombus, 
8 


Cc Cc: D D, 


wi ——_——---1m 


IIl4 ELEMENLALY FID SITIOS, [81 


and let ABDC in the diagram (Fig. 39) represent that cross- 
section. 


If the rhombus ABDC be deformed so as to become | 


ABD,C,, that deformation is a simple shear. It is plain that 
a simple shear is equivalent to an extension in lines parallel to 
AD, and a contraction in those at right angles to AD. The 
directions AD and CB are called the principal axes of the 
shear. The amount of the shear is the displacement of the 
planes per unit of distance from the fixed plane; that is, ste 
is the amount of the shear. 

The stresses that give rise to a simple shear can plainly be 
conceived of as consisting of two equal couples, the forces 
comprising which act tangentially upon parallel planes which 
are moved over one another, and make equal angles with the 
axes of the shear. The forces making up these couples may 
be BOE eo two and two, a and @,a, and 3, (Fig. 40), 

i making up a tension normal to the dimin- 

ished axis; @, and 4, a and @,, making up 

\ a pressure normal to the increased axis. 

N These stresses are measured per unit of area 

Vie of the undeformed sides or sections of the 

seers solid. 

wake? The resistance offered by a body to a 

shearing-stress is called its rigidity, and the ratio of a very 

small change in the stress to the corresponding increment in 
the amount of the shear is called the modulus of rigidity. 

81. Elasticity of Tension.— The first experimental deter- 
iminations of the relations between the elongation of a solid 
and the tension acting on it were made by Hooke in 1678. 
Experimenting with wires of different materials, he found that 
for small tensions the elongation is proportional to the stress. 
It was afterwards found that this law is true for small com- 
pressions. 


32] MOLECULAR MECHANICS. T15 


The ratio of the stress to the elongation of unit length of a 
wire of unit section is the modulus of tracttonal elasticity. For 
different wires it is found that the elongation is proportional 
to the length of the wires, and inversely to their section.’ The 
formula embodying these facts is 


Sl 
ah em 
bs 


(28) 


where ¢ is the elongation, 7 the length, s the section of the 
wire, S the stress, and 4 the modulus of tractional elasticity. 

A method of expressing the modulus of elasticity, due to 
‘Thomas Young, is sometimes valuable. ‘We may express the 
elasticity of any substance by the weight of a certain column 
of the same substance, which may be denominated the modu- 
lus of its elasticity, and of which the weight is such that any 
addition to it would increase it in the same proportion as the 
weight added would shorten, by its pressure, a portion of the 
substance of equal diameter.’’ For example, considering a 
cubic litre of air at o° C. and 7€o0 millimetres of mercury pres- 
sure, and calling its weight unity, we find, from the fact that 
the weight of one litre of mercury is 10517 times that of a 
litre of air, that the pressure of the atmosphere upon a square 
decimetre is 79929 units. If we conceive the air as of equal 
density throughout, this pressure is equivalent to the weight 
of a column of air one square decimetre in section and 7992.9 
metres high. The weight of this column is the modulus of 
elasticity for air; for we know, by Boyle’s law, that if the 
column be altered in length, and its weight therefore cor- 
respondingly altered, the volume of the cubic litre of air 
under consideration will also alter inversely. The height of 
such a column of air as we have assumed is called the height 
of the homogeneous atmosphere. 

82. Elasticity of Torsion—When a cylindrical wire, 
clamped at one end, is subjected at the other to the action of a 


116 ELEMENTARY PHYSICS. [82 


couple the axis of which is the axis of the cylinder, it is found 
that the amount of torsion, measured by the angle of displace- 
ment of the arm of the couple, is proportional to the moment 
of the couple, to the length of the wire, and inversely to the 
fourth power of its radius. It also depends on the modulus 
of rigidity. The formulated statement of these facts is 


Clee tats (29) 


where 7 is the amount of torsion, 7 the length, 7 the radius of 
the wire, C the moment of couple, and z the modulus of rigid- 
ity. No general formula can be found for wires with sections 
of variable form. 

The laws of torsion in wires were first investigated by Cou- 
lomb, who applied them in the construction of an apparatus of 
great value for the measurement of small forces. ) 

The apparatus consists essentially of a small cylindrical wire, 
suspended firmly from the centre of a disk, upon which is cut 
a graduated circle. By the rotation of this disk any required 
amount of torsion may be given to the wire. On the other 
extremity of the wire is fixed, horizontally, a bar, to the ends 
‘ of which the forces constituting the couple are applied. Ar- 
rangements are also made by which the angular deviation of 
this bar from the point of equilibrium may be determined. 
When forces are applied to the bar, it may be brought back to 
its former point of equilibrium by rotation of the upper disk. 
Let © represent the moment of torsion, that is, the couple 
which, acting on an arm of unit length, will give the wire an 
amount of torsion equal to a radian, C the moment of couple 
acting on the bar, t the amount of torsion measured in ra- 
dians; then 


Oa Or. 


82] MOLECULAR MECHANICS. I17 


We may find the value of © in absolute measure by a method 
of oscillations analogous to that used to determine g with the 
pendulum. 

A body of which the moment of inertia can be determined 
by calculation is substituted for the bar, and the time 7 of one 
of its oscillations about the position of equilibrium observed. 

Since the amount of torsion is proportional to the moment 
of couple, the oscillating body has a simple harmonic motion. 

If a represent the amplitude of oscillation of any particle at 
distance ~ from the axis of rotation, we have a=~rr. The 
velocity of the particle at the point of equilibrium is then 


27a 


—_— ——) 


oi 


and the angular velocity of the body, therefore, equals 


22T 


Tr 


The kinetic energy of a body rotating about a centre is $/"*; 
and the kinetic energy of the body considered, at the point of 
equilibrium, is, therefore, 

174%. 

eo ey fae 

The potential energy due to the torsion of the wire is 307’, 
since $@r is the average moment of couple, and 7 the distance 
through which this couple acts. These expressions are neces- 
sarily equal: hence 


or 


118 ELEMENTARY PHYSICS. [83 


We may use a single instead of a double oscillation, when we 
may write the formula 


ogaed oa 


OR (30) 


This apparatus was used by Coulomb in his investigation of 
the law of electrical and magnetic actions. It was also employed 
by Cavendish, as has been already noticed, to determine the 
constant of gravitation. 3 

83. Elasticity of Flexure.—If a rectangular bar be 
clamped by one end, and acted on at the other by a force 
normal to one of its sides, it will be bent or flexed. The 
amount of flexure—that is, the amount of displacement of the 
extremity of the bar from its original position—is found to be 
proportional to the force, to the cube of the length of the bar, 
and inversely to its breadth, to the cube of its thickness, and 
to the modulus of tractional elasticity. The formula there- 
fore becomes 

4h’ 
DMs (31) 


84. Limits of Elasticity.—The theoretical deductions and 
empirical formulas which we have hitherto been considering are 
strictly applicable only to perfectly elastic bodies. It is found 
that the voluminal elasticity of fluids is perfect, and that within 
certain limits of deformation, varying for different bodies, we 
may consider the elasticity of solids to be practically perfect 
for every kind of strain. Ifthe strain be carried beyond the 
limit of perfect elasticity, the body is permanently deformed. 
This permanent deformation is called sez. 

Upon these facts we may base a distinction between solids. 
and fluids: a so/zd requires the stress acting on it to exceed a 
certain limit before any permanent set occurs, and it makes no 


84] MOLECULAR MECHANICS. II9 


difference how long the stress acts provided it lie within the 
limits. A /éuzd, on the contrary, may be deformed by the 
slightest shearing stress, provided time enough be allowed for 
the movement to take place. The fundamental difference lies 
in the fact that fluids offer no resistance to shearing stress other 
than that due to internal friction or viscosity. 

A solid, if it be deformed by a slight stress, is soft, if only 
by a great stress,is hard or rzgvd. A fluid, if deformed quickly 
by any stress, is mobdzle ; if slowly, is vescous. | 

It must not be understood, however, that the behavior of 
elastic solids under stress is entirely independent of time. If, 
for example, a steel wire be stretched by a weight which is 
nearly, but not quite, sufficient to produce an immediate set, it 
is found that, after some time has elapsed, the wire acquires a 
permanent set. If, on the other hand, a weight be put upon 
the wire somewhat less than is required to break it, by al- 
lowing intervals of time to elapse between the successive ad- 
ditions of small weights, the total weight supported by the 
wire may be raised considerably above the breaking-weight. If 
the weight stretching the wire be removed, the return to its 
original form is not immediate, but gradual. If the wire car- 
rying the weight be twisted, and the weight set oscillating by 
the torsion of the wire, it is found that the oscillations die away 
faster than can be explained by any imperfections in the elas- 
ticity of the wire. 

These and similar phenomena are manifestly dependent 
upon peculiarities of molecular arrangement and motion. The 
last two are exhibitions of the so-called viscosity of solids. 
~The molecules of solids, just as those of liquids, move among 
themselves, but with a certain amount of frictional resistance. 
This resistance causes the external work done by the body to 
be diminished, and the internal work done among the mole- 
cules becomes transformed into heat. 


CH ee Rays 
MECHANICS OF FLUIDS. 


85. Pascal’s Law.—A jerfect flutd may be defined as a 
body which offers no resistance to shearing-stress. No actual 
fluids are perfect. Even those which approximate that condi- 
tion most nearly, offer resistance to shearing-stress, due to 
their viscosity. With most, however, a very short time only 
is needed for this resistance to vanish; and all mobile fluids at 
test can be dealt with as if they were perfect, in determining 
the conditions of equilibrium. If they are in motion, their 
viscosity becomes a more important factor. 

Asa consequence of this definition of a perfect fluid, follows 
a most important deduction. In a fluid in equilibrium, not 
acted on by any outside forces except the pressure of the con- 
taining vessel, the pressure at every point and in every direc- 
tion is the same. This law was first stated by Pascal, and is 
known as Pascal's law. 

The truth of Pascal’s law appears, if, in a fluid fulfilling the 
conditions indicated, we imagine a cube of the fluid to become 
solidified. Then, if the law as just stated were not true, there 
would be an unbalanced force in some direction, and the cube 
would move, which is contrary to the statement that the fluid 
isin equilibrium. Ifa vessel filled with a fluid be fitted with a 
number of pistons of equal area A, and a force Ap be applied 
to one of them, acting inwards, a pressure AP will act outwards 
upon the face of each of the pistons. ‘These pressures may be 
balanced by a force applied to each piston. If 2+ 1 be the 
number of the pistons, the outward pressure on # of them, 
caused by the force applied to one, is “pA. 


86] . MECHANICS OF FLUIDS. ae 


The fluid will be in equilibrium when a pressure p is acting 
on unit area of each piston. It is plain that the same reason- 
ing will hold if the area of one of the pistons be A and of an- 
other be zd. A pressure Af on the one will balance a pres- 
sure of mAp on theother. This principle governs the action of 
the hydrostatic press. 

86. Relations of Fluid Pressures due to Outside Forces. 
—Ilf forces, such as. gravitation, act on the mass of a fluid from 
without, Pascal’s law no longer holds true. For suppose the 
cube of solidified fluid to be acted on by gravity; then the 
pressure on the upper face must be less than that on the 
lower face by the weight of the cube, in order that the fluid 
may still be in equilibrium. As the cube may be made as 
small as we please, it appears that, in the limit, the pressure 
on the two faces only differs by an infinitesimal; that is, the 
pressure ina fluid acted on by outside forces is the same at one 
point for all directions, but varies continuously for different 
points. 

The surface of a fluid of uniform density acted on by grav- 
ity, if at rest, is everywhere perpendicular to the lines of force ; 
for, if this were not so, the force at a point on the surface 
could be resolved into two components, one normal and the 
other tangent to the surface. But, from the nature of a fluid, 
the tangential force would set up a motion of the fluid, which 
is contrary to the statement that the fluid is at rest. Ifa sur- 
face be drawn through the points in the field at which the 
pressure is the same, that surface will be perpendicular to the 
lines of force. For, consider a filament of solidified fluid lying 
in the surface; its two ends suffer equal and opposite pressures ; 
hence, since by hypothesis the fluid is in equilibrium, the force 
acting upon it, due to gravity, can have no component in the 
direction of its length, and is perpendicular to the surface in 
which it lies. 

Surfaces of equal pressures are equipotential surfaces. In 


122 ELEMENTARY PTE SIGS: - [87 


small masses of fluid, in which the lines of force due to gravity 
are parallel, these surfaces are horizontal planes. In larger 
masses, such as the oceans, they are curved to correspond to 
the divergence of the lines of force from the centre of the earth. 

In a liquid the pressure at a point is proportional to its 
depth below the surface-of the liquid. For, imagine two rec- 
tangular prisms of solidified liquid with bases which are equal 
and coincident with the surface of the liquid, and with heights 
such that the one is # times the other. From the fact that 
liquids are practically incompressible, the weight of these 
prisms acting downwards is proportional to their volumes, 
and hence to their heights. Since the liquid is in equilibrium, 
these weights are balanced by the upward pressures on their 
lower bases. These pressures are therefore proportional to 
the heights of the prisms, or to the depths of the surfaces to 
which they are applied. 

From the foregoing principles, it is evident that a liquid 
contained in two communicating vessels of any shape whatever 
will stand at the same level in both. If one, however, be filled 
with a liquid of different density from that in the other, equi- 
librium will be established when the depths are inversely as 
the densities of the liquids. 

87. The Barometer.—The instrument best adapted to il- 
lustrate these principles, and also of great importance in many 
physical investigations, is the barometer. It was invented by 
Torricelli, a pupil of Galileo. The fact that water can be raised 
in a tube in which acomplete or partial vacuum has been made 
was known to the ancients, and was explained by them, and by 
the schoolmen after them, by the maxim that ‘ Nature abhors 
a vacuum. They must have been familiar with the action of 
pumps, for the force-pump, a far more complicated instrument, 
was invented by Ctesibius of Alexandria, who lived during the 
second century B.c. It was not until the time of Galileo, 
however, that the first recorded observations were made that 


87] MACHANICS OF FLUIDS. 123 


the column of water in a pump rises only to a height of about 
10.5 metres. Galileo failed to give the true explanation of this 
fact. He had, however, taught that the air has weight; and 
his pupil Torricelli, using that principle, was more successful. 

He showed, that if a glass tube sealed at one end, over 760 
millimetres long, were filled with mercury, the open end stopped 
with the finger, the tube inverted, and the unsealed end plunged 
beneath a surface of mercury in a basin, on withdrawing the 
finger the mercury in the tube sank until its top surface was 
about 760 millimetres above the surface of the mercury in the 
basin. The specific gravity of the mercury being 13.59, the 
weight of the mercury column and that of the water column in 
the pump agreed so nearly as to show that the maintenance of 
the columns in both cases was due to a common cause,—the 
pressure of the atmosphere. This conclusion was subsequently 
verified and established by Pascal, who requested a friend to 
observe the height of the mercury column at the bottom and 
at the top of a mountain. On making the observation, the 
height of the column at the top was found to be less than at 
the bottom. Pascal himself afterwards observed a slight though 
distinct diminution in the height of the column on ascending 
the tower of St. Jacques de la Boucherie in Paris. 

The form of barometer first made by Torricelli is still often 
used, especially when the instrument is stationary, and is in- 
tended to be one of precision. In the finest instruments of 
this class a tube is used which is three or four centimetres in 
diameter, so as to avoid the correction for capillarity. A screw 
of known length, pointed at both ends, is arranged so as to 
move vertically above the surface of the mercury in the cistern. 
When an observation is to be made, the screw is moved until 
its lower point just touches the surface. The distance between 
its upper point and the top of the column is measured by 
means of a cathetometer; and this distance added to the 
length of the screw gives the height of the column. 


124 ELEMENTARY PHYSICS. [88 


Other forms of the instrument are used, most of which are 
arranged with reference to convenient transportability. Vari- 
ous contrivances are added by means of which the column is 
made to move an index, and thus record the pressure on a 
graduated scale. All these forms are only modifications of 
Torricelli’s original instrument. 

The pressure indicated by the barometer is usually stated 
in terms of the height of the column. Mercury being practi- 
cally incompressible, this height is manifestly proportional to 
the pressure at any point in the surface of the mercury in the 
cistern. The pressure on any given area in that surface can be 
calculated if we know the value of g at the place and the spe- 
cific gravity of mercury, as well as the height of the column. 
The standard barometric pressure, represented by 760 millime- 
tres of mercury, is a pressure of 1.033 kilograms on every 
square centimetre. It is called a pressure of one atmosphere ; 
and pressures are often measured by atmospheres. 

In the preparation of an accurate barometer, it is necessary 
that all air be removed from the mercury: otherwise it will 
collect in the upper part of the tube, by its pressure lower the 
top of the column, and make the barometer read too low. 
The air is removed by partially filling the tube with mercury, 
which is then boiled in the tube, gradually adding small quan- 
tities of mercury, and boiling after each addition, until the 
tube is filled. The boiling must not be carried too far; for 
there is danger, in this process, of expelling the air so com- 
pletely that the mercury will adhere to the sides of the tube, 
and will not move freely. For rough work the tube may be 
filled with cold mercury, and the air removed by gently tap. 
ping the tube, so inclining it that the small bubbles of air 
which form can coalesce, and finally be set free at the surface 
of the mercury. 

88. Archimedes’ Principle—If a solid be immersed in a 
fluid, it loses in weight an amount equal to the weight of the 


go] WMWECHANICS OF FLUIDS, 125 


fluid displaced. This law is known, from its discoverer, as 
Archimedes principle. 

The truth of this law will appear if we consider the space 
occupied by the solid as filled with the fluid. The fluid in this 
space will then be in equilibrium, and the upward pressure on 
it must exceed the downward pressure by an amount equal to 
its weight. The resultant of the pressure acts through the 
centre of gravity of the assumed portion of fluid, otherwise 
equilibrium would not exist. If, now, the solid occupy the 
space, the difference between the upward and the downward 
pressures on it must still be the same as before,—namely, the 
weight of the fluid displaced by the solid; that is, the solid 
loses in apparent weight an amount equal to the weight of the 
displaced fluid. 

89. Floating Bodies——When the solid floats on the fluid, 
the weight of the solid is balanced by the upward pressure. 
In order that the solid shall be in equilibrium, these forces 
must act in thesameline. The resultant of the pressure, which 
lies in the vertical line passing through the centre of gravity of 
the displaced fluid, must pass through the centre of gravity of 
the solid. Draw the line in the solid joining these two centres, 
and call it the arzs of the solid. The equilibrium is stable 
when, for any infinitesimal inclination of the axis from the ver- 
tical, the vertical line of upward pressure cuts the axis in a 
point above the centre of gravity of the solid. This point is 
called the metacentre. 

90. Specific Gravity.—Archimedes’ principle is used to de- 
termine the specific gravity of bodies. The specific gravity of 
a body is defined as the ratio of its weight to the weight of an 
equal volume of pure water at a standard temperature. 

The specific gravity of a solid that is not acted on by water 
may be determined by means of the hydrostatic balance. The 
body under examination, if it will sink in water, is suspended 
from one scale-pan of a balance by a fine thread, and is weighed. 


126 ELEMENTARY PHYSICS. [90 


{ft is then immersed in water, and is weighed again. The 
difference between the weights in air and in water is the weight 
of the displaced water, and the ratio of the weight of the body 
to the weight of the displaced water is the specific gravity of 
the body. | 

If the body will not sink in water, a sinker of unknown 
weight and specific gravity is suspended from the balance, and 
counterpoised in water. Then the body, the specific gravity of 
which is sought, is attached to the sinker, and it is found that 
the equilibrium is destroyed. To restore it, weights must be 
added to the same side. ‘These, being added to the weight of 
the body, represent the weight of the water displaced. 

The specific gravity of a liquid is obtained by first balancing 
in air a mass of some solid, such as platinum or glass, that is 
not acted on chemically by the liquid, and then immersing the 
mass successively in the liquid to be tested and in water. The 
ratio of the weights which must be used to restore equilibrium 
in each case is the specific gravity of the liquid. , 

The specific gravity of a liquid may also be found by means 
of the specific gravity bottle. This is a bottle fitted with a 
ground glass stopper. The weight of the water which com- 
pletely fills it is determined once for all. When the specific 
gravity of any liquid is desired, the bottle is filled with the 
liquid, and the weight of the liquid determined. The ratio of 
this weight to the weight of an equal volume of water is the 
specific gravity of the liquid. 

The same bottle may be used to determine the specific 
gravity of any solid which cannot be obtained in continuous 
masses, but is friable or granular. A weighed amount of the 
solid is introduced into the bottle, which is then filled with 
water, and the weight of the joint contents of the bottle deter- 
mined. The difference between the last weight and the sum 
of the weights of the solid and of the water filling the bottle 
is the weight of the water displaced by the solid. The ratio 


90] MECHANICS OF FLUIDS. 127 


of the weight of the solid to the weight thus obtained is the 
specific gravity of the solid. 

The specific gravity of a liquid may also be obtained by 
means of hydrometers. Vhese are of two kinds,—the hydro- 
meters of constant weight and those of constani volume. The 
first consists usually of a glass bulb surmounted by a cylindri- 
cal stem. The bulb is weighted, so as to sink in pure water to 
some definite point on the stem. This point is taken as the 
zero; and, by successive trials with different liquids of known 
specific gravity, points are found on the stem to which the 
hydrometer sinks in these liquids. With these as a basis, 
the divisions of the scale are determined and cut on the stem. 

The hydrometer of constant volume consists of a bulb 
weighted so as to stand upright in the liquid, bearing on the 
top of a narrow stem a small pan, in which weights may be 
placed: The weight of the hydrometer being known, it is im- 
mersed in water; and, by the addition of weights in the pan, 
a fixed point on the stem is brought to coincide with the sur- 
face of the water. The instrument is then transferred to the 
liquid to be tested, and the weights in the pan changed until 
the fixed point again comes to the surface of the liquid. The 
sum of the weight of the hydrometer and the weights added 
in each case gives the weight of equal volumes of water and 
of the liquid, from which the specific gravity sought is easily 
obtained. 

The specific gravity of gases is often referred to air or to 
hydrogen instead of water. It is best determined by filling a 
large glass flask, of known weight, with the gas, the specific 
gravity of which is to be obtained, and weighing it, noting the 
temperature and the pressure of the gas in the flask. The 
weight of the gas at the standard temperature and pressure is 
then calculated, and the ratio of this weight to the weight of 
the same volume of the standard gas is the specific gravity 
desired. The weight of the flask used in this experiment must 


128 ELEMENTARY PHYSICS. [or 


be very exactly determined. The presence of the air vitiates 
all weighings performed in it, by diminishing the true weight 
of the body to be weighed and of the weights employed, by 
an amount proportional to their volumes. The consequent 
error is avoided either by performing the weighings in a 
vacuum produced by the air-pump, or by correcting the appar- 
ent weight in air to the true weight. Knowing the specific 
gravity of the weights and of the body to be weighed, and the 
specific gravity of air, this can easily be done. 

g1. Motions of Fluids.—lIf the parts of the fluid be mov- 
ing relatively to each other or to its bounding-surface, the cir- 
cumstances of the motion can be determined only by making 
limitations which are not actually found in nature. There 
thus arise certain definitions to which we assume that the fluid 
under consideration conforms. 

The motion of a fluid is said to be uxzform when each ele- 
ment of it has the same velocity at all points of its path. The 
motion is steady when, at any one point, the velocity and 
direction of motion of the elements successively arriving at 
that point remain the same for each element. If either the 
velocity or direction of motion change for successive elements, 
the motion is said to be varyzmg. The motion is further said 
to be rotational or trrotational according as the elements of 
the fluid have or have not an angular velocity about their 
axes. 

In all discussions of the motions of fluids a condition is 
supposed to hold, called the condztion of continuity. It is as- 
sumed that, in any volume selected in the fluid, the change of 
density in that volume depends solely on the difference between 
the amounts of fluid flowing into and out of that volume. In 
an incompressible fluid, or liquid, if the influx be reckoned plus. 
and the efflux minus, we have, letting Q represent the amount 
of the liquid passing through the boundary in any one direc- 
tion, 2Q =o. The results obtained in the discussion of fluid. 


Q2) MECHANICS OF FLUIDS. 129 


ee 


motions must all be interpreted consistently with this condition. 
If the motion be such that the fluid breaks up into discontinu- 
ous parts, any results obtained by hydrodynamical considera- 
ations no longer hold true. 

If we consider any stream of incompressible fluid, of which 
the cross-sections at two points where the velocities of the ele- 
ments are v, and uv, have respectively the areas A, and A,, we 
can deduce at once from the condition of continuity 


Vai y is rons TM (32) 


92. Velocity of Efflux.—We shall now apply this principle 
to discover the velocity of efflux of a liquid from an orifice in 
the walls of a vessel. 

Consider any small portion of the 
liquid, bounded by stream lines, which we Nae alas } 
may calla filament. Represent the velocity = 
of the filament at B (Fig. 41) by v,, and at h 
C by v, and the areas of the cross-sections 
of the elements at the same points by A, 
and A. We have then, as above, 4,v, = Av. 
We assume that the flow has been estab- 
lished for a time sufficiently long for the motion to become 
steady. The energy of the mass contained in the filament be- 
tween Band C is, therefore, constant. Let V, represent the 
‘potential at B due to gravity, V the potential at C, and d the 
density of the liquid. The mass that enters at Bina unit of 
time is 


BiGaar. 


aA,v,. 


The mass that goes out at C is the equal quantity dAv. The 
energy entering at B is 


aA Aes a Vy; 


130 ELEMENTARY (PAY SICSS yy [92 


the energy passing out at C is 
aAv(sv" + V). , 
If the pressures at B and C on unit areas be expressed by 
p, and g, the work done at & on the entering mass by the 
pressure 7, is ~,A,v,, and at C on the outgoing mass is pAv. 


The energy within the filament remaining constant, the incom- 
ing must equal the outgoing energy; therefore 


pAv + dAv(av + V) = p,A,v, + dA,v,40; + V,), 
whence, since 4,v, = Av, we have 


FtwtVaS4 iy t+V, 


We may write this equation 


Ke — 2) =(%.—V) ESF; G33) 


Oreavain, since 4.7) — vig, 


wh )=(%-n1) 4454 Gy 


To apply equation (34) to the case of a liquid flowing freely 
into air from an orifice at C, we observe that the difference of 
potential (V, — V) equals the work done in carrying a gram 
from C to B or equals giz — #,), where % represents the height 


92] MECHANICS OF FLUIDS. | I3I 


of the surface above C, and &#, that of the surface above B. 
Further, we have 


Pp; = pa + agh,, 


where , is the atmospheric pressure. At the orifice p equals 
foe ve have then 


w(t — 71) = 84 —h) + gh, = gh, 


whence 


If, now, A become indefinitely small as compared with A,, in 
the limit, the velocity at C becomes 


v= V2gh; (35) 


that is, the velocity of efflux of a small stream issuing from an 
orifice in the wall of a vessel is independent of the density of 
the liquid, and is equal to the velocity which a body would 
-acquire in falling freely through a distance equal to that 
between the surface of the liquid and the orifice. 

This theorem was first given by Torricelli from consid- 
erations based on experiment, and is known as Torrtcelli’s 
theorem. 

We may apply the general equation to the case of the 
efflux of a liquid through a siphon. A szphon is a bent tube 
which is used to convey a liquid by its own weight over a 
barrier. One end of the siphon is immersed in the liquid, and 


132 ELEMENTARY PV SITCS. [92 


the discharging end, which must be below the level of the 
liquid, opens on the other side of the barrier. To set the 
siphon in operation it must be first filled with the liquid, after 


which a steady flow commences. 
2 


rae 
both = ~,, and (V,— V) = gl, where 7 is the distance be- 
tween the surface level and the discharging orifice. The ve- 


In this case, as before, we may set = 0, U, = Ope 


locity becomes v = ¥ 2g/7. The siphon, therefore, discharges. 
more rapidly the greater the distance between the surface level 
and the orifice. It is manifest that the height of the bend in 
the tube cannot be greater than that at which atmospheric 
pressure would support the liquid. 

The flow of a liquid into the vacuum formed in the tube of 
an ordinary pump may also be discussed by the same equation. 
The Sump consists essentially of a tube, fitted near the bottom 
with a partition, in which is a valve opening upwards, In the 
tube slides a tightly fitting piston, in which is a valve, also 
opening upwards. ‘The piston is first driven down to the par- 
tition in the tube, and the enclosed air escapes through the 
valve in the piston. When the piston 1s raised, the liquid in 
which the lower end of the tube is immersed passes through 
the valve in the partition, rises in the tube and fills the space 
left behind the piston. When the piston is again lowered, 
tlre space above it is filled with the liquid, which is lifted out 
of the tube at the next up-stroke. 

To determine the velocity of the liquid following the pis- 
ton, we notice that in this case ~, = pf, and p = Oo if the piston 
move upward very rapidly, (V,.— V) = — gh, where & is the 


height of the top of the liquid column above the free surface 
2 


in the reservoir, and qe again =o. We thenmuave 
1 


92] MECHANICS OF FLUIDS. 133 


The velocity when 4 = ois 


U— 


ap, 
a 


When £ is such that dg# = g,., v = O, which expresses the con- 
dition of equilibrium. 


The equation v= ye expresses, more generally, the ve- 


locity of efflux, through a small orifice, of any fluid of density 
@, from a region in which it is under a constant pressure Z,, into 
a vacuum. 

Torricelli’s theorem is shown to be approximately true by 
allowing liquids to run from an orifice in the side of a vessel, 
and measuring the path of the stream. If the theorem be 
true, this ought to be a parabola, of which the intersection of 
the plane of the stream and of the surface of the liquid is the 
directrix ; for each portion of the liquid, after it has passed the 
orifice, will behave asa solid body, and move in a parabolic 
path. The equation of this path is found, as in § 44, to be 


2u" i 
—_—-A=). 


& 


Now by Torricelli’s theorem, we may substitute for wv’ its 
value 2gh, whence y =4hx. In this equation, since the initial 
movement of the stream is supposed to be horizontal, the per- 
pendicular line through the orifice being the axis of the para- 
bola, and the orifice being the origin, & is the distance from 
the orifice to the directrix. Experiments of this kind have 
been frequently tried, and the results found to approximate 
more nearly to the theoretical as various causes of error were 
removed. 

When, however, we attempt to calculate the amount of 


134 ELEMENTARY PHYSICS. [93 


liquid discharged in a given time, there is found to be a wider 
discrepancy between the results of calculation and the ob- 
served facts. Newton first noticed that the diameter of the 
jet at a short distance from the orifice is less than that of the 
orifice. He'showed this to be a consequence of the freedom 
of motion among the particles in the vessel. The particles 
flow from all directions towards the orifice, those moving from 
the sides necessarily issuing in streams inclined towards the 
axis of the jet. Newton showed that by taking the diameter 
of the narrow part of the jet, which is called the vena contracta, 
as the diameter of the orifice, the calculated amount of liquid 
escaping agreed far more closely with theory. 

When the orifice is fitted with a short cylindrical tube, the 
interference of the different particles of the liquid is in some 
degree lessened, and the quantity discharged increases nearly 
to that required by theory. 

93. Diminution of Pressure.—The Sprengel air-pump, an 
important piece of apparatus to be described hereafter, de- 
pends for its operation on the diminution 
of pressure at points along the line of a 
flowing column of liquid. Let us con- 
sider a large reservoir filled with liquid, 
which runs from it by a vertical tube en- 
tering the bottom of the reservoir. From 
Eq. 34 the value of J, the pressure at any 
point in the tube, is 


p=2,4+(V,—V)d— tae’(1 — 45) 


The ratio qi may be set equal to zero. 
ENaC, If # (Fig. 42) represent the height of the 
upper surface above the point in the tube at which we desire 


to find the pressure, then (V, —- V)=gh. . We then have 


94] MECHANICS OF FLUIDS. 135 


P=h.t+ deh —tdv°. “If the tube be always filled with the 
liquid, Av = A,v,, where A and A, represent the areas of the 
cross-sections of the tube at the point we are considering and 
at the bottom of the tube, and vw and wv, represent the corre- 
sponding velocities. Further, v,° = 2g, if #, represent the 
distance from the upper surface to the bottom of the tube. 
We obtain, by substitution, 


[ Fate 
p= Pa + dg\h — “Fs hs). (36) 


If h equal ae we have p= ,; and if an opening be 


made in the wall of the tube, the moving liquid and the air will 
be in equilibrium. If % be less than fs 7s the pressure / will 
be less than f,, and air will flow into the tube. Since this ine- 
quality exists when A, = 4, it follows, that, if a liquid flow 
from a reservoir down a cylindrical tube, the pressure at any 
point in the wall of the tube is less than the atmospheric pres- 
sure by an amount equal to the pressure of a-column of the 
liquid, the height of which is equal to the distance between 
the point considered and the bottom of the tube. 

94. Vortices.—A series of most interesting results has 
been obtained by Helmholtz, Thomson, and others, from the 
discussion of the rotational motions of fluids. Though the 
proofs are of such a nature that they cannot be presented 
here, the results are so important that they will be briefly 


stated. 
A vortex line is defined as the line which coincides at every 


point with the instantaneous axis of rotation of the fluid ele- 
ment at that point. A vortex filament is any portion of the 
fluid bounded by vortex lines. 

A vortex is a vortex filament which has “contiguous to it 
over its whole boundary irrotationally moving fluid.” 


136 ELEMENTARY PHYSICS. 194 


The theorems relating to this form of motion, as first proved 
by Helmholtz, in 1868, show that,— 

(1) A vortex in a perfect fluid always contains the same 
fluid elements, no matter what its motion through the sur- 
rounding fluid may be. 

(2) The strength of a vortex, which is the product of its 
angular velocity by its cross-section, is constant; therefore the 
vortex in an infinite fluid must always bea closed curve, which, 
however, may be knotted and twisted in any way whatever. 

(3) Ina finite fluid the vortex may be open, its two ends 
terminating in the surface of the fluid. 

(4) The irrotationally moving fluid around a vortex has a 
motion due to its presence, and transmits the influence of the 
motion of one vortex to another. 

(5) If the vortices considered be infinitely long and recti- 
linear, any one of them, if alone in the fluid, will remain fixed 
in position. 

(6) If two such vortices be present parallel to one another, 
they revolve about their common centre of gravity. 

(7) If the vortices be circular, any one of them, if alone, 
moves with a constant velocity along its axis, at right angles 
to the plane of the circle, in the direction of the motion of the 
fluid rotating on the inner surface of the ring. 

(8) The fluid encircled by the ring moves along its axis in 
the direction of the motion of the ring, and with a greater 
velocity. 

(9) If two circular vortices move along the same axis, one 
following the other, the one in the rear moves faster, and 
diminishes in diameter; the one in advance moves slower, and 
increases in diameter. If the strength and size of the two be 
nearly equal, the one in the rear overtakes the other, and 
passes through it. The two now having changed places, the 
action is repeated indefinitely. 

(10) If two circular vortices of equal strength move along 


95] MECHANICS OF FLUIDS. 137 


the same axis toward one another, the velocities of both grad- 
ually decrease and their diameters increase. The same result 
follows if one such vortex move toward a solid barrier. 

The preceding statements apply only to vortices set up in 
a perfect fluid. They may, however, be illustrated by experi- 
ment. To produce circular vortices in the air, we use a box 
which has one of its ends flexible. A circular opening is cut 
in the opposite end. The box is filled with smoke or with 
finely divided sal-ammoniac, resulting from the combination of 
the vapors of ammonia and hydrochloric acid. On striking the 
flexible end of the box, smoke rings are at once sent out. 

The smoke ring is easily seen to be made up of particles re- 
volving about a central core in the form of a ring. With such 
rings many of the preceding statements may be verified. 

An illustration of the open vortex is seen when an oar- 
blade is drawn through the water. By making such open vor- 
tices, using a circular disk, many of the observations with the 
smoke-rings may be repeated in another form. 

95. Air-Pumps.—The fact that gases, unlike liquids, are 
easily compressed, and obey Boyle’s law under ordinary condi- 
tions of temperature and pressure, underlies the construction 
and operation of several pieces of apparatus employed in phy- 
sical investigations. The most important of these is the azr- 
pump. 

The working portion of the air-pump is constructed essen- 
tially like the common lifting-pump already described. The 
valves must be light and accurately fitted. The vessel from 
which the air is to be exhausted is joined to the pump bya 
tube, the orifice of which is closed by the valve in the bottom 
of the cylinder. 

A special form of vessel much used in connection with the 
air-pump is called the vecezver. It is usually a glass cylinder, 
open at one end, and closed bya hemispherical portion at the 
other. The edge of the cylinder at the open end is ground 


138 ELEMENTARY PHYSICS. L95 


perfectly true, so that all points in it are in the same plane. 
This ground edge fits upon a plane surface of roughened brass, 
or ground glass, called the p/aze, through which enters the tube 
which joins the receiver to the cylinder of the pump. The 
joint between the receiver and the plate is made tight by a 
little oil or vaseline. 

The action of the pump is as follows: as the piston is 
raised, the pressure on the upper surface of the valve in the 
cylinder is diminished, and the air in the vessel expands in ac- 
cordance with Boyle’s law, lifts the valve, and distributes itself 
in the cylinder, so that the pressure at all points in the vessel 
and the cylinder is the same. The piston is now forced down, 
the lower valve is closed by the increased pressure on its upper 
surface, the valve in the piston is opened, and the air in the 
cylinder escapes. At each successive stroke of the pump this 
process is repeated, until the pressure of the remnant of air left 
in the vessel is no longer sufficient to lift the valves. 

The density of the air left in the vessel after a given num- 
ber of strokes is determined, provided there be no leakage, by 
the relations of the volumes of the vessel and the cylinder. 

Let V represent the volume of the vessel, and C that of the 
cylinder when the piston is raised to the full extent of the 
stroke. Let d and d, respectively represent the density of the. 
air in the vessel before and after one stroke has been made. 
After one down and one up stroke have been made, the air 


which filled the volume V now fills V+ C. It follows that 


d, V 


i Sa 


As this ratio is constant no matter what density may be con- 
sidered, it follows that, if @, represent the density after x 
strokes, 


Slee) (37) 


95] MECHANICS OF FLUIDS. 139 


As this fraction cannot vanish until 2 becomes infinite, it is 
plain that a perfect vacuum can never, even theoretically, be 
obtained by means of the air-pump. If, however, the cylinder 
be large, the fraction decreases rapidly, and a few strokes are 
sufficient to bring the density to such a point that either the 
pressure is insufficient to lift the valves, or the leakage through 
the various joints of the pump counterbalances the effect of 
longer pumping. 

In the best air-pumps the valves are made to open auto- 


matically. In Fig. 43 is represented one of the methods by 
which this is accomplished. They can then be made heavier 
and with a larger surface of contact, so that the leakage is di- 
minished, and the limit of the useful action of the pump is 
much extended. With the best pumps of this sort a pressure 
of one-half a millimetre of mercury is reached. 

The Sprengel air-pump depends for its action upon the 
principle, discussed in § 93, that a stream of liquid running 
down acylinder diminishes the pressure upon its walls. In the 


440 ELEMENTARY PHYSICS. [96 


Sprengel pump the liquid used is mercury. It runs from a 
large vessel down a glass tube, into the wall of which, at a dis- 
tance from the bottom of the tube of more than 760 milli- 
metres, enters the tube which connects with the receiver. The 
lower end of the vertical tube dips into mercury, which pre- 
vents air from passing up along the walls of the tube. When 
the stream of mercury first begins to flow, the air enters the 
column from the receiver, in consequence of the diminished 
pressure, passes down with the mercury in large bubbles, and 
emerges at the bottom of the tube. As the exhaustion pro- 
ceeds, the bubbles become smaller and less frequent, and the 
mercury falls in the tube with a sharp, metallic sound. It is 
evident that, as in the case of the ordinary air-pump, a perfect 
vacuum cannot be secured. There is no leakage, however, in 
this form of the air-pump, and a very high degree of exhaus- 
tion can be reached. 

The Morren or Alvergniat mercury-pump is in principle 
merely a common air-pump, in which combinations of stop- 
cocks are used instead of valves, and a column of mercury in 
place of the piston. Its particular excellence is that there is 
scarcely any leakage. 

The compressing-pump is used, as its name implies, to in- 
crease the density of air or any other gas within a receiver. 
The receiver in this case is generally a strong metallic vessel. 
The working parts of the pump are precisely those of the air- 
pump, with the exception that the valves open downwards, 
As the piston is raised, air enters the cylinder, and is forced 
into the receiver at the down-stroke. 

96. Manometers.—The manometer is an instrument used 
for measuring pressures. One variety depends for its opera- 
ation upon the regularity of change of volume of a gas with 
change of pressure. This, in its typical form, consists of a 
heavy glass tube of uniform bore, sealed at one end, with the 
open end immersed in a basin of mercury. The pressure to be 


98] MECHANICS OF FLOIDS. LAF 


measured is applied to the surface of the mercury in the basin. 
As this pressure increases, the air contained in the tube is 
compressed, and a column of mercury is forced up the tube. 
The top of this column serves as an index. We know, from 
Boyle’s law, that, when the volume of the air has diminished 
one-half, the pressure is doubled. The downward pressure of 
the mercury column makes up a part of this pressure; and the 
pressure acting on the surface of the mercury in the basin is 
greater than that indicated by the compression of the air in the 
tube, by the pressure due to the mercury column. For many 
purposes the manometer tube may be made very short, and 
the pressure of the mercury column that rises in it may be 
neglected. 

97. Aneroids.—The aucroid is an instrument used to de- 
termine ordinary atmospheric pressures. On account both of 
its delicacy and its easy transportability, it is, often used in- 
stead of the barometer. It consists of a metallic box, the 
eover of which is made of thin sheet-metal corrugated in cir- 
cular grooves. The air is partially exhausted from the box, 
and it is then sealed. Any change in the pressure of the at- 
mosphere causes the corrugated top to move. This motion is 
very slight, but is made perceptible, either by a combination of 
levers, which amplifies it, or by an arm rigidly fixed on the 
top, the motion of which is observed by a microscope. The 
indications of the aneroid are compared with those of a stand- 
ard mercurial barometer, and an empirical scale is thus made, 
by means of which the aneroid may be used to determine 
pressures directly. 

98. Limitations to the Accuracy of Boyle’s Law.—In 
all the previous discussions, we have dealt with gases as if they 
obeyed Boyle’s law with absolute exactness. This, however, is 
not the case. In the first place, some gases at ordinary tem- 
peratures can be liquefied by pressure. As these gases ap- 
proach more nearly the point of liquefaction, the product py 


142 ELEMENTARY PHYSICS. [98 


of the volume and pressure becomes less than it ought to be 
in accordance with Boyle’s law. 

Secondly, those gases which cannot be liquefied at ordinary 
temperatures by any pressure, however great, show a different 
departure from the law. For every gas, except hydrogen, 
there is a minimum value of the product pv. At ordinary 
temperatures and small pressures the gas follows Boyle's law 


quite closely, becoming, however, more compressible as the 


pressure increases, until the minimum value of fy is reached. 
It then becomes gradually less compressible, and at high pres- 
sures its volume is much greater than that determined by 
Boyle’s law. If the temperature be raised, the agreement with 
the law is closer, and the pressure at which the minimum value 
of pv occurs is greater. Hydrogen seems to differ from the 
other gases, only in that the pressures at which the observa- 
tions upon it were made were probably greater than the one at 
which its minimum value of py occurs. The volume of the 
compressed hydrogen is uniformly greater than that required 
by Boyle’s law. 

Important modifications are introduced into the behavior 
of gases under pressure by subjecting them to intense cold. 
It is then found that all gases, ype exception, can be lique- 
fied, and even solidified. 

The subject is intimately connected with the subject of 
critical temperature, and will be again discussed under Heat. 


HEAT. 


CLAP LE RT: 
MEASUREMENT OF HEAT. 


99. General Effects of Heat.—Bodies are warmed, or 
their temperature is raised, by heat. The sense of touch is 
often sufficient to show difference in temperature; but the true 
criterion is the transfer of heat from the hotter to the colder 
body when the two bodies are brought in contact, and no work 
is done by one upon the other. This transfer is known by 
some of the effects described below. 

Bodies, in general, expand when heated. Experiment shows 
that different substances expand differently for the same rise 
of temperature. Gases, in general, expand more than liquids, 
and liquids more than solids, Expansion, however, does not 
universally accompany rise of temperature. A few substances 
contract when heated. 

Heat changes the state of aggregation of bodies, always in 
such a way as to admit of greater freedom of motion among 
the molecules. Vhe melting of ice and the conversion of water 
into steam are familiar examples. 

Heat breaks up chemical compounds. The compounds of 
sodium, potassium, lithium, and other metals, give to the flame 
of a Bunsen lamp the characteristic colors of the vapors of the 
metals which they contain. This fact shows that the heat 
separates the metals from their combinations. 


144 ELEMENTARY) PAN siGa: [100 


When the junction of two dissimilar metals in a conducting 
circuit is heated, electric currents are produced. 

Heat performs mechanical work. For example, the heat 
produced in the furnace of a steam-boiler may be used to drive 
an engine. 

100. Production of Heat.—Heat is produced by various 
processes, some of which are the reverse of the operations just 
mentioned as the effects of heat. As examples of such reverse 
operations may be mentioned, the production of heat by the 
compression of a body which expands when heated; the pro- 
duction of heat during a change in the state of aggregation of 
a body, when the freedom of motion among the molecules is 
diminished; the production of heat during chemical combina- 
tion; and the production-of heat when an electric current 
passes through a junction of two dissimilar metals in an oppo- 
site direction to that of the current which is set up when the 
junction is heated. 

Heat is produced in general in any process involving the 
expenditure of mechanical energy. The heat produced in such 
processes cannot be used to restore the whole of the original 
mechanical energy. The production of heat by friction is the 
best example of these processes. 

Further, an electric current, in a homogeneous conductor, 
generates heat at every point in it, while, if every point in the 
conductor be equally heated, no current will be set up. 

These cases are examples of the production of heat by non- 
reversible processes. 

101. Nature of Heat.—Heat was formerly considered to 
be a substance which passed from one body to another, lower- 
ing the temperature of the one and raising that of the other, 
which combined with solids to form liquids, and with liquids 
to form gases or vapors. But the most delicate balances fail 
to show any change of weight when heat passes from one body 
to another. Rumford was able to raise a considerable quan- 


101] MEASUREMENT OF HEAT. 145 


tity of water to the boiling-point by the friction of a blunt 
boring-tool within the bore of acannon. He showed that the 
heat manifested in this experiment could not have come from 
any of the bodies present, and also that heat would continue 
to be developed as long as the borer continued to revolve, or 
that the supply of heat was practically inexhaustible. The 
heat, therefore, must have been generated by the friction. 

That ice is not melted by the combination with it of a 
heat substance was shown early in the present century by 
Davy. He caused ice to melt by friction of one piece upon 
another in a vacuum, the experiment being performed in a 
room where the temperature was below the melting-point of 
ice. There was no source from which heat could be drawn. 
The ice must, therefore, have been melted by the friction. 

Rumford was convinced that the heat obtained in his ex- 
periment was only transformed mechanical energy; but to. 
demonstrate this it was necessary to prove that the quantity 
of heat produced was always proportional to the quantity of 
mechanical work done. This was done in the most complete 
manner by Joule in a series of experiments extending from 
1842 to 1849. He showed, that, however the heat was pro- 
duced by mechanical means, whether by the agitation of water 
by a paddle-wheel, the agitation of mercury, or the friction of 
iron plates upon each other, the same expenditure of mechani- 
cal energy always developed the same quantity of heat. Joule 
also proved the perfect equivalence of heat and electrical 
energy. 

These experiments prove that heat 7s a form of energy. 
Consistent explanations of all the phenomena of heat may be 
given if we assume that the molecules of all bodies are in con- 
stant motion, that the temperature of a body varies with the 
mean kinetic energy of its molecules, and that the heat in a 


body is the sum of the kinetic energies of its molecules. 
pHe) 


149 ELEMEN TAR YIP Y SLC [ac2 


THERMOMETRY. 


102. Temperature.—Two bodies'are said to be at the 
same lemperature when, if they be brought into intimate con- 
tact, no heat is transferred from one to the other. A body is 
at a high temperature relatively to other bodies when it gives 
up heat to them. The fact that it gives up heat may be shown 
by its change in volume. A body is at a low temperature 
when it receives heat from surrounding bodies. It is under- 
stood, of course, in what is said above, that one body has no 
action upon the other; in other words, no work is done by 
one body upon the other when they are brought in contact. 

103. Thermometers.— Experiments show that, in general, 
bodies expand, and their temperature rises progressively, with 
the application of heat. An instrument may be constructed 
which will show at any instant the volume of a body selected 
for the purpose. If the volume increase, we know that the 
temperature rises; if the volume remain constant or diminish, 
we know that the temperature remains stationary or falls. 
Such an instrument is called a thermometer. 

The thermometer most in use consists of a glass bulb with 
a fine tube attached. The bulb and part of the tube contain 
mercury. In order that the thermometers of different mak- 
ers may give similar readings, it is necessary to adopt two 
standard temperatures which can be easily and certainly re- 
produced. The temperatures adopted are the melting-point 
of ice, and the temperature of steam from boiling water, under 
a pressure equal to that of a column of mercury 760 millime- 
tres high at Paris. After the instrument has been filled with 
mercury, it is plunged in melting ice, from which the water is 
allowed to drain away, and a mark is made upon the stem op- 
posite the end of the mercury column. It is then placed in a 
vessel in which water is boiled, so constructed that the steam 
rises through a tube surrounding the thermometer, and then 


103] MEASUREMENT OF HEAT. 147 


descends by an annular space between that tube and an outer 
one, and escapes at the bottom. The thermometer does not 
touch the water, but is entirely surrounded by steam. The 
point reached by the end of the mercury column is marked on 
the stem, as before. The space between these two marks is 
then divided into a number of equal parts. 

While all makers of thermometers have adopted the same 
standard temperatures for the fixed points of the scale, they dif- 
fer as to the number of divisions between these points. The 
thermometers used for scientific purposes, and in general use in 
France, have the space between the fixed points divided into 
a hundred equal parts or degrees. ‘The melting-point of ice is 
marked o°, and the boiling-point 100°. This scale is called 
the Centigrade or Celsius scale. 

The Aéaumur scale, in use in Germany, has eighty degrees 
between the melting- and boiling-points, and the boiling-point 
is marked 80°. 

The Fahrenhezt scale, in general use in England and Amer- 
ica, has a hundred and eighty degrees between the melting- 
and boiling-points. The former is marked 32°, and the latter 
A beds 

The divisions in all these cases are extended below the zero 
point, and are numbered from zero downward. ‘Temperatures 
below zero must, therefore, be read and treated as negative 
quantities. 

A few points in the process of construction of a thermom- 
eter deserve notice. . It is found that glass, after it has been 
heated to a high temperature and again cooled, does not for 
some time return to its original volume. The bulb of a ther- 
mometer must be heated in the process of filling with mercury, 
and it will not return to its normal volume for some months. 
The constructiou of the scale should not be proceeded with un. 
til the reservoir has ceased to contract. For the same reason, 
if the thermometer be used for high temperatures, even the 


148 ELEMENTARY PHYSICS. [104 


temperature of boiling water, time must be given for the reser- 
voir to return to its original volume before it is used for the 
measurement of low temperatures. 

It is essential that the diameter of the tube should be nearly 
uniform throughout, and that the divisions of the scale should 
represent equal capacities in the tube. To test the tube a 
thread of mercury about 50 millimetres long is introduced, and 
its length is measured in different parts of the tube. If the 
length vary by more than a millimetre, the tube should be re- 
jected. If the tube be found to be suitable, a bulb is attached, 
mercury is introduced, and the tube sealed after the mercury 
has been heated to expel the air. When it is ready for gradu- 
ation, the fixed points are determined; then a thread of mer- 
cury having a length equal to about ten degrees of the scale is. 
detached from the column, and its length measured in all parts. 
of the tube. By reference to these measurements, the tube is 
so graduated that the divisions represent parts of equal capac- 
ity, and are not necessarily of equal length. 

If such a thermometer indicate a temperature of 10°, this. 
means that the thermometer is in such a thermal condition that 
the volume of the mercury has increased from zero one tenth 
of its total expansion from zero to 100°. There is no reason 
for supposing that this represents the same proportional rise of - 
temperature. Ifa thermometer be constructed in the manner 
described, using some liquid other than mercury, it will not 
in general indicate the same temperature as the mercurial ther- 
mometer, except at the two standard points. It is plain, there- 
fore, that a given fraction of the expansion of a liquid from 
zero to 100° cannot be taken as representing the same fraction 
of the rise of temperature. 

104. Air Thermometer.—If a gas be heated, and its vol- 
ume kept constant, its pressure increases. For all the so- 
called permanent gases—that is, those which are liquefied only 
with great difficulty—the amount of increase in pressure for 


105] MEASUREMENT OF HEAT. 149 


the same increase of temperature is found to be almost ex. 
actly the same. This fact is a reason for supposing that the 
increase of pressure is proportional to the increase of tempera- 
ture. There are theoretical reasons, as will be seen later, for 
the same supposition. 

An instrument constructed to take advantage of this in- 
rease in pressure to measure temperature is called an azr ther- 
mometer. A bulb so arranged that it may be placed in the 
medium of which the temperature is to be determined, is filled 
with air or some other gas, and means are provided for main- 
taining the volume of the gas constant, and measuring its pres- 
sure. For the reasons given above, the air thermometer is 
taken as the standard instrument for scientific purposes. Its 
use, however, involves several careful observations and tedious 
computations. It is, therefore, mainly employed as a standard 
with which to compare other instruments. If we make sucha 
comparison, and construct a table of corrections, we may re- 
duce the readings of any thermometer to the corresponding 
readings of the air thermometer. 

105. Limits inthe Range of the Mercurial Thermom- 
eter.— The range of temperature for which the mercurial ther- 
mometer may be employed is limited by the freezing of the 
mercury on the one hand, and its boiling on the other. For 
temperatures below the freezing-point of mercury, alcohol 
thermometers may be employed. For the measurement of 
high temperatures, several different methods have been em- 
ployed. One depends upon the expansion of a bar of plati- 
num, another upon the variation in the electric resistance of 
platinum wire, another upon the strength of the electric cur- 
rent generated by,a thermo-electric pair, another on the density 
of mercury vapor. This Jast method is carried out as follows: 
A globe of refractory material, fitted with a short tube with a 
small bore, contains a small quantity of mercury. It is placed 
in the furnace or other place, the temperature of which is to 


150 ELEMENTARY PHYSICS. [106 


be measured. The mercury boils, the air and the excess of 
mercury are expelled, and the globe is finally left full of mer- 
cury vapor at the temperature of the furnace. The globe is 
cooled, and the weight of the mercury left in it determined. 
From this and the volume of the globe the temperature can 
be computed. 

106. Registering Thermometers.—MVaxtmum and mint- 
mum thermometers are employed to register the highest and 
lowest temperatures reached during a given period. By a 
change in construction, the ordinary mercury thermometer be- 
comes a self registering maximum thermometer. This change 
consists in making a contraction in the tube just above the 
reservoir, to such an extent that, though the mercury is pushed 
through it as the temperature rises, it does not return as the 
temperature falls. It thus serves as an index to show the high- 
est temperature reached during the period of its exposure. 
After an observation has been made, the thermometer is re- 
adjusted for a new observation by allowing the instrument to 
swing out of the horizontal position in which it usually rests, 
about a point near the upper extremity of the tube. 

In the construction of the minimum thermometer, alcohol 
is the liquid employed. - Before sealing, an index of glass, 
smaller in diameter than the bore of the tube, is inserted. 
When the instrument is adjusted for use, this is brought in 
contact with the extremity of the column, and the tube is 
placed in a horizontal position. If, now, the alcohol expand, 
‘it will flow past the index without moving it; but if it con- 
tract, it will, by adhesion, draw the index after it. The mini- 
mum temperature is thus registered. 

Registering thermometers have been made to give a con- 
tinuous record of changes of temperature. One method of 
effecting this is to produce an image of the thermometer tube, 
which is strongly illuminated by a light placed behind it, upon 
a screen of sensitized paper which moves continuously by means 


108| MEASUREMENT OF HEAT. I5!I 


of clockwork. Light is excluded from the whole of the paper, 
except the part that corresponds to the image of the tube 
above the mercury. This part of the paper is blackened by 
the light; and, as the paper moves, the edge of the blackened 
portion will present a sinuous line corresponding to the move- 
ments of the mercury of the thermometer. 


CALORIMETRY. 


107. Unit of Heat.—It is evident that more heat is required 
to raise the temperature of a large quantity of a substance 
through a given number of degrees than to raise the tempera- 
ture of a small quantity of the same substance through the 
same number of degrees. It is further evident that the suc- 
cessive repetition of any operation by which heat is produced 
will generate more heat than a single operation. Heat is 
therefore a quantity the magnitude of which may be expressed 
in terms of some unit. The unit of heat generally adopted is 
the heat required to raise the temperature of one kilogram of 
water from zero to one degree. It is called a calorze. 

It is sometimes convenient to employ a smaller unit, name- 
ly, the quantity of heat necessary to raise one gram of water 
from zero to one degree. This unit is designated as the lesser 
calorie. It is one one-thousandth of the larger unit. It may, 
therefore, be called a szzl/icalorte. 

The fact that heat is energy enables us to employ still 
another unit. It is that quantity of heat which is equivalent 
toanerg. This unit is called the mechanical unit of heat. A 
calorie contains about 41,595,000,000 mechanical units. 

108. Heat. required to raise the Temperature of a 
Mass of Water.—It is evident that to raise the temperature 
of m kilograms of water from zero to one degree will require 
m calories. If the temperature of the same quantity of water 
fall from one degree to zero, the same quantity of heat is given 
to surrounding bodies. 


152 ELEMENTARY PHYSICS. [109 


Experiment shows, that, if the same quantity of water be 
raised to different temperatures, quantities of heat nearly pro- 
portional to the rise in temperature will be required: hence, 
to raise the temperature of m kilograms of water from zero to 
t degrees requires m¢ calories very nearly. This is shown by 
mixing water at a lower temperature with water at a higher 
temperature. The temperature of the mixture will be almost 
exactly the mean of the two. Regnault, who tried this experi- 
ment with the greatest care, found the temperature of the 
mixture a little higher than the mean, and concluded that the 
quantity of heat required to raise the temperature of a kilo- 
gram of water one degree increases slightly with the tempera- 
ture; that is, to raise the temperature of a kilogram of water 
from twenty to twenty-one degrees, requires a little more heat 
than to raise the temperature of the same quantity of water 
from zero to one degree. 

Rowland found, by mixing water at various temperatures, 
and also by measuring the energy required to raise the tem- 
perature of water by agitation by a paddle-wheel, that, when 
the air thermometer is taken as a standard, the quantity of 
heat necessary to raise the temperature of a given quantity of 
water one degree diminishes slightly from zero to thirty de- 
grees, and then increases to the boiling-point. 

109. Specific Heat.—Only one thirtieth as much heat is 
required to raise the temperature of a kilogram of mercury 
from zero to one degree as is required to raise the temperature 
of a kilogram of water through the same range. In order to 
raise the temperatures of other substances through the same 
range, quantities of heat peculiar to each substance are required. 

The quantity of heat required to raise the temperature of 
one kilogram of a substance from zero to one degree is called 
the specific heat of the substance. 

If the temperature of one kilogram of a substance rise from 
z, to ¢, the limit of the ratio of the quantity of heat required to 


r10] MEASUREMENT OF HEAT. 153 


bring about the rise in temperature to the difference in tem- 
perature, as that difference diminishes indefinitely, is called the 
specific heat of the substance at temperature ¢. If we represent 
Onna 


a iat 


the quantity of heat by Q, the limit of the ratio 


expresses this specific heat. 

The specific heats of substances are generally nearly con- 
stant between zero and one hundred degrees. The mean spe- 
cific heat of a substance between zero and one hundred degrees 
is the one usually given in the tables. 

The measurement of specific heat is one of the important 
objects of calorimetry. 

110. Ice Calorimeter.—JSlack's ice calorimeter consists of 
a block of pure ice having a cavity in its interior covered by a 
thick slab of ice. The body of which the specific heat is to 
be determined is heated to ¢ degrees, then dropped into the 
cavity, and immediately covered by the slab. After a short 
time the temperature of the body falls to zero, and in so doing 
converts a certain quantity of ice into water. This water is 
removed bya sponge of known weight, and its weight is deter- 
mined. It-will be shown, that to melt a kilogram of ice re- 
quires 80 calories; if, then, the weight of the body be P, and 
its specific heat 2, it gives up, in falling from ¢ degrees to 
zero, Prt calories. On the other hand, if g kilograms of ice 
be melted, the heat required is 80f. Therefore Prt = 80) ; 
whence | 


aga pe (38) 


Bunsen's ice calorimeter (Fig. 44) is used for determining 
the specific heats of substances of which only a small quantity 
is at hand. The apparatus is entirely of glass. The tube B is 
filled with water and mercury, the latter extending into the 
graduated capillary tube C. To use the apparatus, alcohol 


154 ELEMENTARY fit YS C in [110 


which has been artificially cooled to a temperature below zero 
is passed through the tube A. A layer of ice forms around 
the outside of this tube. As water 
freezes, it expands. This causes the 
mercury to advance in the capillary 
tube C. When a sufficient quantity 
of ice has been formed, the alcohol 
is removed from A, the apparatus is 
surrounded by melting snow or ice, 
and a small quantity of water is in- 
troduced, which soon falls in tem- 
perature to zero. The position of the 
mercury in C is now noted; and the 
substance the specific heat of which 

li is to be determined, at the tempera- 
ture of the surrounding air, is dropped into the water in A. 
Its temperature quickly falls to zero, and the heat which it 
loses is entirely employed in melting the ice which surrounds 
the tube A. As the ice melts, the mercury in the tube C re- 
treats. The change of position is an indication of the quantity 
of ice melted, and the quantity of ice melted measures the 
heat given up by the substance. The number of divisions of 
the tube C corresponding to one calorie can be determined by 
direct experiment. To make this determination, the opera- 
tion is performed as described above, using a substance of a 
known specific heat c. If its mass be g and its temperature f, 
it gives up, in cooling to zero, cf¢ calories. If the mercury 
retreat at the same time z# divisions, one division corresponds 


cpt 


to —calories. If, now, a mass 7’ of a substance at a tempera-- 
nt 


ature z’ be introduced, and the mercury fall 2’ divisions, the 

number of calories which must have been given to the ice is 

ncpt 
n 


, and the specific heat of the substance is 
n' cht 
A = iid 
np t 


(39) 


111] MEASUREMENT OF HEAT, 155 


11z. Method of Mixtures.—The method of mixtures con- 
sists in bringing together, at different temperatures, the sub- 
stance of which the specific heat is desired and another of which 
the specific heat is known, and noting the change of tempera- 
ture which each undergoes. 

The water calorimeter consists of a vessel of very thin copper 
or brass, highly polished, and placed within another vessel upon 
non-conducting supports. A mass Pof the substance of which 
the specific heat is to be determined ts brought to a tempera- 
ture ¢ in a suitable bath, then plunged in water at the tem- 
perature Z, contained in the calorimeter. The whole will soon 
come to a common temperature @. 

The substance loses Px (¢,— @) calories. 

The heat gained by the calorimeter consists of the heat 
gained by the water, #(9@— 7); the heat gained by the vessel, 
poe (@—2); the heat gained by the glass stirrer and the glass 
of the thermometer, pc’ (6 — 2); the heat gained by the mer- 
cury of the thermometer f’’’c’” (9 —?¢): where # represents the 
mass of the water, #’ the mass of the vessel, #’’ the mass of the 
glass of the stirrer and of the thermometer, #’”’ the mass of the 
mercury of the thermometer, c’ the specific heat of the material 
of the vessel, c’’ the specific heat of glass, and ¢’” the specific 
heat of mercury. If no heat be lost or gained by radiation, the 
heat lost by the substance is equal to that gained by the calori- 
meter: 


whence Pr(t,—0)=(p+pe+pie’ +p" )s(G—2). (40) 


To determine x from this formula, c’, c’’, and c’’’ must be 
known. Approximate values of these may be obtained and 
used in the formula, but it is better to determine the value of 
potp'c’+p'"c'" =p, by experiment. Let amass of water, 
P, at atemperature ¢, be substituted for the substance of which 
the specific heat was to be determined in the experiment de- 
scribed above. The equation will then become 


Ptt,— #=(p+ (9 — 4), (41) 


156 ELEMENTARY PHYSICS. [112 


in which g is the only unknown quantity. /, is the water 
equivalent of the calorimeter and accessories. It is determined, 
once for all, as just described. 

There is a source of error in the use of the instrument, due 
to the radiation of heat during the experiment. This error 
may be nearly eliminated, by making a preliminary experiment 
to determine what change of temperature the calorimeter will 
experience; then, for the final experiment, the calorimeter and 
its contents are brought to a temperature below the tempera- 
ture of the surrounding air, by about half the amount of that 
change. The calorimeter will then receive heat from the sur- 
rounding medium during the first part of the experiment, and 
lose heat during the second part. The rise of temperature is, 
however, much more rapid at the beginning than at the end of 
the experiment. The rise from the initial temperature to the 
temperature of the surrounding medium occupies less time than 
the rise from the latter to the final temperature. The gain of 
heat, therefore, does not exactly compensate for the loss. If 
greater accuracy be required, the rate of cooling of the calori- 
meter must be determined by putting into it warm water, the 
same in quantity as would be used in experiments for deter- 
mining specific heat, and noting its temperature from minute 
to minute. Such an experiment furnishes the data for com-., 
puting the loss or gain by radiation. ‘To secure accurate re- 
sults the body must be transferred from the bath to the calori- 
meter without sensible loss of heat. 

112. Method of Comparison.—The method of comparison 
consists in conveying to the substance of which the specific 
heat is to be determined a known quantity of heat, and com- 
paring the consequent rise of temperature with that produced 
by the same amount of heat in a substance of which the speci- 
fic heat is known. In the early attempts to use this method, 
the heat produced by the same flame burning for a given time 
was applied successively to different liquids. A more exact 


113] MEASUREMENT OF HEAT. 157 


method was the combustion, within the calorimeter, of a known 
weight of hydrogen. The best method of obtaining a known 
quantity of heat is by means of an electrical current of known 
strength flowing through a wire of known resistance wrapped 
upon the calorimeter. 

113. Method of Cooling.—The method of cooling consists. 
in noting the time required for the calorimeter, ina space kept 
constantly at zero, to cool from a temperature 7’ to 
a temperature 7, when empty; when containing a 
given weight of water; and when containing a given 
weight of the substance of which the specific heat 
is sought. The thermo calorimeter of Regnault, rep- 
resented in Fig. 45,is an example. It consists of an 
alcohol thermometer, with its bulb 4 enlarged and 
made in the form of a hollow cylinder, inside of 
which the substance is placed. The thermometer is 
warmed, and then placed in a vessel surrounded by 
melting ice. It radiates heat to the sides of the ves- 
sel, and the column of alcohol in the tube falls. Let 
«x be the time occupied in falling from the division z 
to the division z’ when the space Bis empty. Let 
the times occupied in falling between the same two 
divisions, when the space 4 contains a mass P of 
water, and when it contains a mass P’ of the sub- 
stance of which the specific heat c’ is sought, be re- Fie. 4s. 
spectively x and x". Let JZ be the water equivalent of the 
instrument. We then have 


717 NN Bin wet eed ss 
BEAT Vilrothes Sac! athe 


since, under the conditions of the experiment, the heat lost per 
second must be the same in each case. 


158 Hi ELEMENTARY PHYSICS. [x54 


Eliminating J7, we obtain 
men Sa ee 


114. Determination of the Mechanical Equivalent of 
Heat.—It has been stated that whenever heat is produced by 
the expenditure of mechanical energy, the quantity of heat pro- 
duced is always proportional to the quantity of mechanical 
energy expended. 

The mechanical equivalent of heat is the energy in mechan- 
ical units, the expenditure of which produces the unit of heat. 

Heat applied to a body may increase the motion of its 
molecules;. that is, add to their kinetic energy. It may per- 
form internal work by moving the molecules against molecular 
forces. It may perform external work by producing motion 
against external forces. If we could estimate these effects in 
mechanical units, we might obtain the mechanical equivalent 
of heat. But the kinetic energy of the molecules cannot be 
estimated, for we do not know their mass nor their velocity. 
We must, therefore, in the present state of our knowledge, 
resort to direct experiment to determine the heat equivalent. 
In one of the experiments of Joule, already referred to, a pad- 
dle-wheel was made to revolve, by means of weights, in a vessel 
filled with water. In this vessel were stationary wings, to pre- 
vent the water from acquiring a rotary motion with the paddle- 
wheel. By the revolution of the wheel the water was warmed. 
The heat so generated was estimated from the rise of tempera- 
ture, while the mechanical energy required to produce it was 
given by the fall of the driving-weight. Joule repeated this 
experiment, substituting mercury for the water. In another 
experiment he substituted an iron plate for the paddle-wheel, 
and made it revolve with friction upon a fixed iron plate under 
water. 


114] . MEASUREMENT OF HEAT; I59 


Joule expressed his results in kilogram-metres—that is, 
the work done by a kilogram in falling under the force of 
gravity through one metre. He stated the mechanical equiva- 
lent of one calorie, in this unit, to be 423.9, from the experi- 
ments with water; 425.7, from those with mercury; and 426.1, 
from those with iron plates. He gave the preference to the 


Fic. 46. 


smallest value, and it has been generally accepted as the 
mechanical equivalent. This mechanical equivalent is called 
Joule’s equivalent, and is represented by /. In absolute units it 
is about 41,595,000,000 ergs per calorie. 

Rowland has repeated Joule’s experiment with water; but 
he caused the paddle-wheel to revolve by means of an engine, 
and determined the moment of the couple required to prevent 


160 ELEMENTARY PHYSICS. [114 


the revolution of the calorimeter. Fig. 46 shows the appara- 
tus. The shaft of the paddle-wheel projects through the bot- 
tom of the calorimeter, and is driven by means of a bevel-gear. 
The vessel A is suspended from C by a torsion wire, and its 
tendency to rotate balanced by weights attached to cords 
which act upon the circumference of a pulley D. By this dis- 
position of the apparatus he was able to expend about one half 
a horse-power in the calorimeter, and obtain a rise of tempera- 
ture of 35° per hour; while in Joule’s experiments the rise of 
temperature per hour was less than 1°. These experiments 
give, for the mechanical equivalent of one calorie at 5°, 429.8 
kilogram-metres; at 20°, 426.4 kilogram-metres. 

Several other methods have been employed for determining 
the mechanical equivalent. The concordance of the results by 
all these methods is sufficient to warrant the statement that 
the expenditure of a given amount of mechanical energy 
always produces the same amount of heat. 


CHAPTER TT: 
TRANSFER OF HEAT. 


115. Transfer of Heat.—In the preceding discussions it 
has been assumed that heat may be transferred from one body 
to another, and that if two bodies in contact be at different 
temperatures, heat will be transferred from the hotter to the 
colder body. In general, if transfer of heat be possible in any 
system, heat will pass from the hotter to the colder parts of 
the system, and the temperature of the system will tend to 
become uniform. There are three ways in which this transfer 
is accomplished, called respectively convection, conduction, and 
radiation. 

116. Convection.—If a vessel containing any fluid be heated 
-at the bottom, the bottom layers become less dense than those 
above, producing a condition of instability. The lighter 
portions of the fluid rise, and the heavier portions from above, 
coming to the bottom, are in their turn heated. Hence con- 
tinuous currents are caused. This process is called convection. 
By this process, masses of fluid, although fluids are poor con- 
ductors, may be rapidly heated. Water is often heated ina 
reservoir at a distance from the source of heat by the circula- 
tion produced in pipes leading to the source of heat and back. 
The winds and the great currents of the ocean are convection 
currents. An interesting result follows from the fact that 
water has a maximum density ($135). When the water of lakes 
cools in winter, currents are set up and maintained, so long as 
the surface water becomes more dense by cooling, or until the 


whole mass reaches 4°. Any further cooling makes the 
II 


162 ELEMENTARY vPA YSICS, [117 


surface water lighter. It therefore remains at the surface, and 
its temperature rapidly falls to the freezing-point, while the great 
mass of the water remains at the temperature of its maximum 
density. 

117. Conduction.—If one end of a metal rod be heated, it 
is found that the heat travels along the rod, since those 
portions at a distance from the source of heat finally become 
warm. ‘This process of transfer of heat from molecule to mole- 


cule of a body, while the molecules themselves retain their . 


relative places, is called conduction. 

In the discussion of the transfer of heat by conduction it is 
assumed as a principle, borne out by experiment, that the flow 
of heat between two very near parallel planes, drawn in a sub- 
stance, is proportional to the difference of temperature be- 
tween those planes. 

118. Flow of Heat across a Wall.—To study the transfer 
of heat by conduction, we will consider what takes place ina 
wall of homogeneous material, the exposed surfaces of which, 
assumed to be of indefinite extent, are maintained at a con- 


stant difference of temperature. Suppose the wall to be cut 


by a series of planes parallel to the exposed surfaces; and that 
the state of the body, as respects temperature, has become per- 
manent. Then we will show that there must be the same flow 
of heat across all parallel sections, and also that there must be 


a uniform fall of temperature from one side of the wall to the. 


other,—that is, that if ¢’/—¢ represent the difference of temper- 
ature between the two exposed surfaces, and d the thickness 


/ 


a 


of the wall, is the fall of temperature per unit thickness, 


Hibs 
and 7’ — - 7 
warmer surface. 

To demonstrate this, suppose A (Fig. 47) to be one exposed 


BEY, : 
ad’ is the temperature at a distance @ from the 


— — 


119] TRANSFER OF HEAT. 163 


surface at temperature ¢’, and # the other surface at temper- 
ature ¢: suppose a, a’, a’’, to be three surfaces 
parallel to the faces of the wall, and at very small 
equal distances from one another. Suppose the 
temperatures to exist according to the law stated 
in the proposition: then the difference of temper- 
ature between a and a’ will be the same as between 
@ anda”. It has been stated that the flow of heat 
between two points in a body is proportional to the 
difference of temperature between those points. 
Experiment shows that it depends also upon the 
distance between them, the nature of the material, 
and to a very limited extent upon the temperature 
itself. The effect of this last factor may, however, be neg- 
lected, since the pairs of surface considered are nearly at the 
same temperature. The other factors being the same for 
both pairs, it follows that there will be the same flow of heat 
from a@ to a’ as from a’ to a’. The same will hold true 
for any other set of surfaces parallel to the faces of the wall: 
hence the molecules in any surface such as a’ receive and part 
with equal amounts of heat, and can neither rise nor fall in 
temperature. If the temperatures, therefore, were .once estab- 
lished in accordance with the law enunciated, they could never 
change. On the other hand, if the difference of temperature 
between a@ and a@ were greater than that between a’ and a”,. 
the molecules in a would receive more heat than they would 
part with, their temperature would rise, and this would tend 
to equalize these differences. The proposition is therefore 
demonstrated 

119. Flow Proportional to Rate of Fall of Tempera- 
ture.—It can be further shown that the flow of heat across 
walls of the same material is directly proportional to the 
differences of temperature between the faces of the walls, and 
inversely proportional to the thicknesses of the walls. Repre- 


rr 


A B 


> 
SS 


—----—-~—----— ——----------,-~-—-g 
ee Se a SS See aap amen aoe any eee mrs -------- 


neephiphed leet oe Bega ap ieee ee 


Fic. 47. 


104 ELEMENTARY PHYSICS. [120 


sent the thicknesses of two walls A and B by d and oO respec- 
tively, the temperatures of the exposed surfaces of A by # and 
t,and of B by & and & Assume two planes in each wall parallel 
to the exposed surfaces, at very small equal distances apart, and 
similarly situated. The flow of heat in each wall, from the one 
plane to the other, will be proportional only to the differ- 
ences of temperature, since all other things are equal. If eé 


E i . 
be the common distance between the planes, qe — #) will be 
the difference of temperature between the planes in A, and 


=(0' — @) will be the same in B; hence we have 


cl 
fiow of heat between the planesin A _ a? —?) Sages 
flow of heat between the planesin B = oe 
O 


30-9) 


This proves the proposition, since the heat which flows across: 
the wall is the same as that which flows between any two 


planes. 


It will be seen that d 


— f£ [ 
Tens the vate of fall of temperature 


at the section considered; and it follows, finally, that the flow 
of heat across any section parallel to the exposed surfaces of a 
wall is proportional to the rate of fall of temperature at that 
section. 

120. Conductivity.—If,; now, we consider a prism extend- 
ing across the wall, bounded by planes perpendicular to the 
exposed surfaces, and represent the area of its exposed bases 
by A, the quantity of heat which flows in a time 7 through 
this prism may be represented by 


t’—t 
Q= K—_AT, (43) 


122] TRANSFER OF HEAT. 165 


where XK is a constant depending upon the material of which 
the wall is composed. & is the conductivity of the substance, 
and may be defined as the quantity of heat which in unit time 
flows through a section of unit area in a wall of the substance 
whose thickness is unity, when its exposed surfaces are main- 
tained at a difference of temperature of one degree; or, in 
other words, it is the quantity of heat which in unit time flows 
through a section of unit area in a substance, where the rate of 
fall of temperature at that section is unity. In the above dis- 
cussions the temperatures Z’ and ¢ are taken as the actual tem- 
peratures of the surfaces of the wall. If the colder surface of 
the wall be exposed to air of temperature Z, to which the heat 
which traverses it is given up, ¢ will be greater than 7. The 
difference will depend upon the quantity of heat which flows, 
and upon the facility with which the surface parts with heat. 

121. Flow of Heat along a Bar.—If a prism of a sub- 
stance have one of its bases maintained at a temperature f, 
while the other base and the sides are exposed to air at a 
lower temperature, the conditions of uniform fall of tempera- 
ture no longer exist, and the amount of heat which flows 
through the different sections is no longer the same; but the 
amount of heat which flows through any section is still pro- 
portional to the rate of fall of temperature at that section, and 
is equal to the heat which escapes from the portion of the bar 
beyond the section. 

122. Measurement of Conductivity.—A bar heated at one 
end furnishes a convenient means 
of measuring conductivity. In Fig. 
48 let AB represent a bar heated at 
A: Let the ordinates aa’, 00’, cc’, 
represent the excess of temperatures 
above the temperature of the air at 
the points from which they are 
drawn. These temperatures may be determined by means of 


166 ELEMENTARY PHYSICS. [122 


thermometers inserted in cavities in the bar, or by means of a 
thermopile. Draw the curve a’d’c'd’ ... through the summits. 
of the ordinates. The inclination of this curve at any point 
represents the rate of fall of temperature at that point. 
The ordinates to the line 6m, drawn tangent to the curve at 
the point 0’, show what would be the temperatures at various 
points of the bar if the fall were uniform and at the same rate 
as at 6’. It shows that, at the rate of fall at 0’, the bar would 
at # be at the temperature of the air; or, in the length dm, 
the fall of temperature would equal the amount represented 


/ 


bb 
by 00’. The rate of fall is, therefore, pu If Q represent the 


quantity of heat passing the section at 0 in the unit time, we 
have, from § 120, 


O=K x rate of fall of temperature X area of section. 


Q is equal to the quantity of heat that escapes in unit time 
from all that portion of the bar beyond 4. It may be found 
by heating a short piece of the same bar to a high temperature, 
allowing it to cool under the same conditions that surround 
the bar AZ, and observing its temperature from minute to 
minute as it falls. These observations furnish the data for 
computing the quantity of heat which escapes per minute from 
unit length of the bar at different temperatures. It is then 
easy to compute the amount of heat that escapes per minute 
from each portion, dc, cd, etc., of the bar beyond 6; each portion 
being taken so short that its temperature throughout may, 
without sensible error, be considered uniform and the same as 
that at its middle point. Summing up all these quantities, we 
obtain the quantity Q which passes the section 4 in the unit 
time. Then 


Q 


{Cx ee 
rate of fall of temperature at b Xarea of section 


127] TRANSFER OF HEAT. 167 


123. Conductivity diminishes as Temperature rises.— 
By the method described above, Forbes determined the con- 
ductivity of a bar of iron at points at different distances from 
the heated end, and found that the conductivity is not the 
same at all temperatures, but is greater as the temperature is 
lower. 

124. Conductivity of Crystals.—The conductivity of crys- 
tals of the isometric system is the same in all directions, but 
in crystals of the other systems it is not so. Ina crystal of 
Iceland spar the conductivity is greatest in the direction of the 
axis of symmetry, and equal in all directions ina plane at right 
angles to that axis. 

125. Conductivity of Non-homogeneous Solids.—De la 
Rive and De Candolle were the first to show that wood con- 
ducts heat better in the direction of the fibres than at right 
angles to them. Tyndall, by experimenting upon cubes cut 
from wood, has shown that the conductivity has a maximum 
value parallel to the fibres, a minimum value at right angles to 
the fibres and parallel to the annual layers, and a medium 
value at right angles to both fibres and annual layers. Feath- 
ers, fur, and the materials of clothing, are poor conductors be- 
cause of their want of continuity. 

126. Conductivity of Liquids.—The conductivity of liquids 
can be measured, in the same way as that of solids, by noting 
the fall of temperature at various distances from the source of 
heat in a column of liquid heated at the top. Great care must 
be taken in these experiments to avoid errors due to convec- 
tion currents. 

Liquids are generally poor conductors. 

127. Radiation.—We have now considered those cases in 
which there is a transfer of heat between bodies in contact. 
Heat is also transferred between bodies not in contact. This 
is effected by a process called radzatzon, which will be subse- 
quently considered. 


Gh aN 5h os SS RB 


BP EE Cts) (Om BE Als 


SOLIDS AND LIQUIDS. 


128. Expansion of Solids——-When heat is applied to a 
body it increases the kinetic energy of the molecules, and also 
increases the potential energy, by forcing the molecules farther 
apart against their mutual attractions and any external forces 
that may resist expansion. Since the internal work to be done 
when a solid or liquid expands varies greatly for different sub- 
stances, it might be expected that the amount of expansion 
for a given rise of temperature would vary greatly. 

In studying the expansion of solids, we distinguish /ucar 
and voluminal expansion. 

The increase which occurs in the unit length of a substance 
for a rise of temperature from zero to I° C. is called the coe ff- 
cient of lincar expansion. Experiment shows that the expan- 
sion for a rise of temperature of one degree is very nearly con- © 
stant between zero and 100°. , 

Represent by Z, the distance between two points in a 
body at zero, by /; the distance between the same points at the 
temperature ¢, and by @ the coefficient of linear expansion of 
the substance of which the body is composed. 

The increase in the distance /, for a rise of one degree in 
temperature is a/,, for a rise of ¢ degrees az¢/,. Hence we 
have, after a rise in temperature of ¢ degrees, 


, = 41 + at), (44) 


129] EFFECTS OF HEAT. 169 


l, 
and Z, = fe at? 
or approximately, 4=%4(11 — a). 


The binomial 1 + az is called the factor of expansion. 
In the same way, if & represent the coefficient of voluminal 
expansion, the volume of a body at a temperature ¢ will be 


V, = Vit + ko); . (45) 


and if d@ represent density, since density is inversely as vol- 
ume, we have 


d, 


a 


For a homogeneous solid, the coefficient of voluminal ex- 
pansion is three times that of linear expansion; for, if the 
temperature of a cube, with an edge of unit length, be raised 
one degree, the length of its edge becomes I + a, and its vol- 
ume I+ 3@-+ 3a’-+ a*, Since @ is very small, its square and 
cube may be neglected; and the volume of the cube after a 
rise in temperature of one degree is 1+ 3a. 3a is, therefore, 
the coefficient of voluminal expansion. 

129. Measurement of Coefficients of Linear Expansion. 
—Coefficients of linear expansion are measured by comparing 
the lengths, at different temperatures, of a bar of the substance 
the coefficient of which is required, with the length, at constant 
temperature, of another bar. The constant temperature of the 
latter bar is secured by immersing it in melting ice. The bar, 
the coefficient of which is sought may be brought to different 
temperatures by immersing it ina liquid bath; but it is found 
better to place the bar upon the instrument by means of which 


170 LLEUEN TARY PE VSLGs: [130 


the comparisons are to be made, and leave it for several hours 
exposed to the air of the room, which is kept at a constant 
temperature by artificial means. Of course several hours must 
elapse between any two comparisons by this method, and its 
application is restricted to such ranges of temperature as may 
be obtained in occupied rooms; but within this range the ob- 
servations can be made much more accurately than would be 
the case when the bar is immersed ina bath, and it is within 
this range that an accurate knowledge of coefficients of expan- 
sion is of most importance. 

130. Expansion of Liquids.—In studying the expansion 
of a liquid, it is important to distinguish its abso/ute expansion, 
or the real increase in volume, and its apparent expansion, or its 
increase in volume in comparison with that of the containing 
vessel. | 

131. Absolute Expansion of Mercury.—A knowledge of 
the coefficients of expansion of mercury is of the greatest im- 
portance, since mercury is made use of for so many purposes in 
physical research. Regnault has made the most accurate de- 
terminations of these constants. 

To determine the absolute coefficient, the experiment must 
be so made that the expansion of the vessel shall not influence 
the result. 

Regnault’s method consists in comparing the heights of two 
columns of mercury, at different temperatures, which produce 
the same pressure. Two vertical tubes, ad, a’b’ (Fig. 49), are 
connected at the top by a horizontal tube aa’, and at the bot- 
tom by a tube dcda'’c'b’, a part of which is of glass, and shaped 
like an inverted u. The top of the inverted U-tube is connected 
by a tube e with a vessel f, in which air can be maintained at 
any desired pressure. When these tubes are filled with mer- 
cury, it flows freely from one to the other at the top; but the 
flow of mercury between them at the bottom is prevented by 
air imprisoned in the U-tube, while the pressure is transmitted 


132] EFFECTS OF HEAT. I71 


undiminished. The pressure at each end of the column of im- 
prisoned air must, therefore, be the 
same ; and, since a anda’ are connected 
by a horizontal tube, the pressures at 
those points are the same also; hence 
the difference of pressure between 2 
and @ must be equal to the difference 
between @’ and a’; and from § 86 it fol- 
lows that the heights of the columns, 
without regard to the diameters of the 
tubes, producing this difference, are in- 
versely proportional to their densities. 
If, now, one branch be raised to the 
temperature 7, while the other remains 
at zero, the mercury in the U-tube will 
assume different levels. Measuring the 
height of each column from the surface 
of the mercury in the u-tube to the 
horizontal tube at the top, we have, from Eq. (46), if % and /’ 
represent the height of the cold and warm columns respec- 
tively, 


Fic. 49. 


ea a 

Nem AL re gay ry 

hkt =h' —h; (47) 
_H=h 
We (a Be 


132. Apparent Expansion of Mercury.—lIf a glass bulb 
(Fig. 50), furnished with a capillary tube, be filled with mer- 
cury at zero, and heated to a temperature 7, some of the mer- 
cury runs out. The amount which overflows evidently de- 
pends upon the difference of expansion between the mercury 
and the glass. Let P represent the mass of mercury that fills 


172 ELEMENTARY PHYSICS. [132 


the bulb and tube at zero. After heating, there remains in 
the bulb a mass P — gf of mercury, which at zero 
occupies the volume ad. The mass of mercury Z, 
which runs out, would at the same temperature 
fill the remainder of the bulb and tube. Hence 


ie “ay where d represents the 


P 


density of mercury. The volume above 6 equals 7 


The mercury in ad, when heated to the temperature 7, just 


the volume ad equals HS 


Fic. 50. 


1 
fills the tube; and its apparent volume is 7 If « represent 


the coefficient of apparent expansion, 


P P— 
or 
PORES. 
LE pts 


If we know «, the instrument may be used as a thermometer; 
for, suppose it filled at zero, and subjected to an unknown 
temperature 7,, we shall have, if ~, represent the mercury that 
then runs over, 


P= (P—/p,) (1+ 44), (49) 
whence 


Zz Me et 
; (foicoapy JAS 


The instrument is, therefore, called a wezght thermometer. 
The difference between the value «x, found above, and the 


133] EFFECTS OF HEAT. 173 


absolute coefficient & is due to the expansion of the glass. 
And if 2’ be the coefficient for glass, we have 


k'=k—k; 


for, referring again to Fig. 50, the volume of the vessel at zero 


We fe 
is 7 and at the temperature ¢ is re + ‘k't), which, from Eq. 
(48), equals 
P—p 
(1+ «é)(1+ #2). 


The real volume at the temperature ¢ of the mercury re- 
maining in the tube is 


ee + £7). 3 


Hence 


(1+ xt) (1-42) = (+2); 
(I+ «t+ kt + Kk?) = 1-4 €t. 


Since « and &’ are small quantities, their product may be neg- 
lected; hence 


Pi Need PAF (50): 


133. Determination of Voluminal Expansion of Sol- 
ids.—The weight thermometer may be used to determine the 
coefficient of voluminal expansion of solids. For this purpose, 
the solid, of which the volume at zero is known, must be intro- 
duced into the bulb by the glass-blower. If the bulb contain- 
ing the solid be filled with mercury at zero, and afterward 
heated to the temperature ¢, it is evident that the amount of 
mercury that will overflow will depend upon the coefficient of 


174 ELEMENTARY PHYSICS. [134 


expansion of the solid, and upon the coefficient of apparent ex- 
pansion of mercury. If the latter has been determined for the 
kind of glass used, the former can be deduced. By this means 
the coefficients of voluminal expansion of some solids have 
been determined; and the results are found to verify the con- 
clusion, deduced from theory (§ 128), that the voluminal co- 
efficient is three times the linear. 

134. Absolute Expansion of Liquids other than Mer- 
cury.-—The weight thermometer may also serve to determine 
the coefficients of expansion of liquids other than mercury ; for, 
if &’ has been found as described above, the instrument may 
be filled with the liquid the coefficient of which is desired, and 
the apparent expansion of this liquid found exactly as was that 
of mercury. The absolute coefficient’ for the liquid is then the 
sum of the coefficient of apparent expansion and the coefficient 
for the glass. 

135. Expansion of Water.—The use of water as a stand- 
ard with which to compare the densities of other substances 
makes it necessary to know, not merely its mean coefficient of 
expansion, but its actual expansion, degree by degree. This is 
the more important since water expands very irregularly. The 
best determinations of the volumes of water at different tem- 
peratures are those of Matthiessen. The method which he 
employed was to weigh in water a mass of glass of which the 
coefficient of expansion had been previously determined. 

Water contracts, instead of expanding, from 0° to 4°; from 
that temperature to its boiling-point it expands. 

136. Correction Introduced in the Determination of 
Specific Gravity—Water at its maximum density, at 4°, is 
the standard to which are referred the specific gravities of solids 
and liquids. Since it is seldom practicable to make the deter- 
minations at that temperature, corrections must be made as 
follows: 

Let 4 represent the density of a substance at the tempera- 


139] EFFECTS OF HEAT. 176 


ture 2° compared to water at the same temperature. Let 4, rep- 
resent the density of the substance at ¢° compared to water at 4°. 
Then 4; = 4, X d;, where d; represents the density of water 
at ¢°; for, if Wrepresent the mass of the substance, W, the 
mass of an equal volume of water at 2°, and W, the mass of the 


W W, 
same volume of water at 4°, we have 4, = Wo — —. and 
Zz 4 
W W WwW, 
a fe Ww: Whence 4; = W. ax W, are, AL ts Seren 


sent the density of a substance at o°, J, = 41+ £7), where & 
represents the coefficient of voluminal expansion of the sub- 
stance. 

137. Effect of Variation of Temperature upon Specific 

teat.—It has already been seen (§ 109) that the specific heat 

of bodies changes with temperature. With most substances 
the specific heat increases as the temperature rises. 

For example, the true specific heat of the diamond 


SEE STM MMR rs eh! no swe h inven.) 's. Tet as ies 0h CROAT 
SSSI I ae SN ke Cs) 1S, win) a) os) of Sen” na (OV TAED 
RSE ts ee ee. le hey tkel uct), et!) ORION 
OO Ee er ea Cees eee eS es 


138. Effect of Change of Physical State upon Specific 
Heat.—The specific heat of a substance is not the same when 
in the solid as when in liquid state. In the solid state of the 
substance it is generally less than in the liquid. For example: 


Mean Specific Heat 
SSS 


Solid. Liquid. 
COTE MEN Pe G5) es) XO of 6s) )ery can OLS 1.000 
MERIT ule al en (eels 0). ef loses oh OLOSTA 01333 
iene lh Se RR Uren is Roi a! 0.0637 
PR act ets uh Se! ee kere eChOAtd 0.0402 


139. Atomic Heat.—It has been found that the product 
ef the specific heat by the atomic weight of any simple body 


176 ELEMENTARY PHYSICS. [140 


is a constant quantity. This law is known from its discoverers. 
as the law of Dulong and Petit. 

This law may be otherwise stated, thus: that to raise the 
temperature of an atom of any simple substance one degree, 
an amount of heat is required which is the same for all sub- 
stances. 

The experiments of Regnault show that this law may be 
extended to compound bodies; that is, for all compounds of 
similar chemical composition the product of the total chemical 
equivalent by the specific heat is the same. 

The following table will illustrate the law of Dulong and 
Petit. The atomic weights are those given by Clarke. 


Specific Heat Product of Specific 

ELEMENTS, of Atomic Weight. Heat into Atomic 
Equal Weights. Weight. 
LTO: pees Reece O.114 55.9 6,372 
CODDEI wa iis 0.095 63.17 6,001 
Mercury,’ 0 «|. 0.0314 (solid) 199 71 6.128 
Silver, saws 0.057 107.67 6.1397 
Old aa aie 0.0329 196.15 6.453 
Petia (tee as Cals 0.056 117.7 6.591 
EAC Sh ears yeas 0.0314 206.47 6.483 
FAC RT ay amr re ae 0.0955 64.9 6.198 


140. Fusion and Solidification.—When ice at a tempera- 
ture below zero is heated, its temperature rises to zero, and 
then the ice begins to melt; and, however high the tempera- 
ture of the medium that surrounds it may be, its temperature 
remains constant at zero so long as it remains in the solid 
state. This temperature is the melting-point of ice, and because 
of its fixity it is used as one of the standard temperatures in 
eraduating thermometric scales. Other bodies melt at very 
different but at fixed and definite temperatures. Many sub- 
stances cannot be melted, as they decompose by heat. 

Alloys often melt at a lower temperature than any of 
their constituents. An alloy of one part lead, one part tin, four 


/ 


142| BEPECT OF MEAT, ry: 


parts bismuth, melts at 94°; while the lowest melting-point 
of its constituents is that of tin, 228°. An alloy of lead, tin, 
bismuth, and cadmium melts at 62°. 

If a liquid be placed in a medium the temperature of which 
is below its melting-point, it will, in general, begin to solidify 
when its temperature reaches its melting-point, and it will re- 
main at that temperature until it is all solidified. Under cer- 
tain conditions, fiowever, the temperature of a liquid may be 
lowered several degrees below its melting-point without solidi- 
fication, as will be seen below. 

141. Change of Volume with Change of State.—Sub- 
stances are generally more dense in the solid than in the liquid 
state, but there are some notable exceptions. Water, on solidi- 
fying, expands ; so that the density of ice at zero is only 0.9167, 
while that of water at 4° is 1. This expansion exerts consider- 
able force, as is evidenced by the bursting of vessels and pipes 
_containing water. 

142. Change of Melting- and Freezing-Points.—lf water 
be enclosed in a vessel sufficiently strong to prevent its expan- 
sion, it cannot freeze except at a lower temperature. The 
freezing-point of water is, therefore, lowered by pressure. On 

’ the other hand, substances which contract on solidifying have 
their solidification hastened by pressure. 

The lowering of the melting-point of ice by pressure explains 
some remarkable phenomena. If pieces of ice be pressed to- 
gether, even in warm water, they will be firmly united. Frag- 
ments of ice may be moulded, under heavy pressure, into a 
solid, transparent mass. This soldering together of masses of 
ice is called regelation. If a loop of wire be placed over a 
block of ice and weighted, it will cut its way slowly through 
the ice, and regelation will occur behind it. After the wire has 
passed through, the block will be found one solid mass, as 
before. The explanation of these phenomena is, that the ice 


is partially melted by the pressure. The liquid thus formed is 
[2 


178 ELEMENTARY PHYSICS. [143 


colder than the ice; it finds its way to points of less pressure, 
and there, because of its low temperature, it congeals, firmly 
uniting the two masses. 

Water, when freed from air and kept perfectly quiet, will 
not form ice at the ordinary freezing-point. Its temperature 
may be lowered to — 10° or — 12° without solidification. In 
this condition a slight jar, or the introduction of a small frag- 
ment of ice, will cause a sudden congelation of part of the 
liquid, accompanied by a rise in temperature in the whole mass 
LOGZEIO. 


A similar phenomenon is observed in the case of several 


solutions, notably sodium sulphate and sodium acetate. If a 
saturated hot solution of one of these salts be made, and al- 
lowed to cool in a closed bottle in perfect quiet, it will not 
crystallize. Upon opening the bottle and admitting air, crys- 
tallization commences, and spreads rapidly through the mass, 
accompanied by a considerable rise of temperature. If the 
amount of salt dissolved in the water be not too great, the so- 
lution will remain liquid when cooled in the open air, and it 
may even suffer considerable disturbance by foreign bodies 
without crystallization; but crystallization begins immediately 
upon contact with the smallest crystal of the same salt. 

143. Heat Equivalent of Fusion.—Some facts that have 
appeared in the above account of the phenomena of fusion and 
solidification require further study. It has been seen that, 
however rapidly the temperature of a solid may be rising, the 
inoment fusion begins the rise of temperature ceases. What- 
ever the heat to which a solid may be exposed, it cannot be 
made hotter than its melting-point. When ice is melted by 
pressure, its temperature is lowered. When a liquid is cooled, 
its fall of temperature ceases when solidification begins; and 
if, as may occur under favorable conditions, a liquid is cooled 
below its melting-point, its temperature rises at once to the 
melting-point, when solidification begins. Heat, therefore, dis- 


145] EFFECTS OF HEAT. 179 


appears when a body melts, and is generated when a liquid be- 
comes solid. 

It was stated (§ 101) that ice can be melted by friction; 
that is, by the expenditure of mechanical energy. Fusion is, 
therefore, work which requires the expenditure of some form 
of energy to accomplish it. The heat required to melt unit 
mass of a substance is the eat equivalent of fusion of that sub- 
stance. When a substance solidifies, it develops the same 
amount of heat as was required to melt it. 

144. Nature of the Energy stored in the Liquid._-From 
the facts given above, as well as from the principle of the con- 
servation of energy, it appears that the energy expended in 
melting a body is stored in the liquid. It is easy to see what 
must be the nature of this energy. When a body solidifies, its 
molecules assume certain positions in obedience to their mu- 
tual attractions. When it is melted, the molecules are forced 
into new positions in opposition to the attractive forces. They 
are, therefore, in positions of advantage with respect to these 
forces, and possess potential energy. 

145. Determination of the Heat Equivalent of Fusion. 
—The heat equivalent of fusion may be determined by the 
method of mixtures ($111), as follows: a mass of ice, for ex- 
ample, represented by /, at a temperature Z below its melting- 
point, to insure dryness, is plunged into a mass / of warm 
water at the temperature 7. Represent by @ the resulting 
temperature, when the ice is all melted. If p represent the 
water equivalent of the calorimeter, (P’ + p)(Z— @) is the 
heat given up by the calorimeter and its contents. Let c 
represent the specific heat of ice, and x the heat equivalent of 
fusion.» The-ice absorbs, to raise its temperature to zero, Pte 
calories; to melt it, Pr calories; to warm the water after melt- 
ing, P@ calories. We then have the equation 


Pic+ P04 Pe = (P'+p)(T— 9), 


from which x may be found. 


180 ELEMENTARY PHYSICS. [146 


Other calorimetric methods may be employed. The best 
experiments give, for the heat equivalent of fusion of ice, very 
nearly eighty calories. 


GASES AND VAPORS. 


146. The Gaseous State.—A gas may be defined as a 
highly compressible fluid. A given mass of gas has no definite 
volume. Its volume varies with every change in the external 
pressure to which it is exposed. A vapor is the gaseous state 
of a substance which at ordinary temperatures exists as a solid 
or a liquid. 

147. Vaporization is the process of formation of vapor. 
There are two phases of the process, evaporatzon, in which vapor 
is formed at the free surface of the liquid, and edu//etion, in 
which the vapor is formed in bubbles in the mass of the liquid, 
or at the heated surface with which it is in contact. 

148. Nature of the Process of Evaporation.—It has 
been seen (§ 101) that there are many reasons for believing that 
the molecules of solids and liquids are in a state of continual 
motion. It is not supposed that any one molecule maintains 
continuously the same condition of motion; but in the inter- 
action of the molecules the motion of any one may be more or 
less violent, as it receives motion from its neighbors, or gives 
up motion to them. It can easily be supposed that, at the ex- 
posed surface of the substance, the motion of a molecule may 
at times be so violent as to project it beyond the reach of the 
molecular attractions. If this occur in the air, or in a space 
filled with any gas, the molecule may be turned back, and made 
to rejoin the molecules in the liquid mass; but many will find 
their way to such a distance that they will not return. They 
then constitute a vapor of the substance. As the number of 
free molecules in the space above the liquid increases, it is plain 
that there may come a time when as many will rejoin the liquid 
as escape from it. The space is then saturated with the vapor. 


149} EFFECTS OF HEAT. 181 


The more violent the motion in the liquid, that is the higher 
its temperature, the more rapidly the molecules will escape, and 
the greater must be the number in the space above the liquid 
before the returning will equal in number the outgoing mole- 
cules. In other words, the higher the temperature, the more 
dense the vapor that saturates a given space. If the space 
above a liquid be a vacuum, the escaping molecules will at first 
meet with no obstruction, and, as a consequence, the space will 
be very quickly saturated with the vapor. 

Experiment verifies all these deductions. Evaporation goes 
on continually from the free surfaces of many liquids, and even 
of solids. It increases in rapidity as the temperature increases, 
and ceases when the vapor has reached a certain density, 
always the same for the same temperature, but greater for a 
higher temperature. It goes on very rapidly in a vacuum; but 
it is found that the final density of the vapor is no greater, or 
but little greater, than when some other gas is present. In 
other words, while a foreign gas impedes the motion of the 
outgoing molecules, and causes evaporation to go on slowly, it 
has very little influence upon the number of molecules that 
must be present in order that those which return may equal 
in number those which escape. 

149. Pressure of Vapors.—As a liquid evaporates in a 
closed space, the vapor formed exerts a pressure upon the en- 
closure and upon the surface of the liquid, which increases so 
long as the quantity of vapor increases, and reaches a maxi- 
mum when the space is saturated. This maximum pressure of 
a vapor increases with the temperature. When evaporation 
takes place in a space filled by another gas which has no action 
upon the vapor, the pressure of the vapor is added to that of 
the gas, and the pressure of the mixture is, therefore, the sum 
of the pressures of its constituents. The law was announced 
by Dalton that the quantity of vapor which saturates a given 
‘space, and consequently the maximum pressure of that vapor, 
is the same whether the space be empty or contain a gas. 


182 ELEMENTARY PHYSICS. [150 


Regnault has shown that, for water, ether, and some other 
substances, the maximum pressure of their vapors is slightly 
less when air is present. 

150. Ebullition.—As the temperature of a liquid rises, the 
pressure which its vapor may exert increases, until a point is 
reached where the vapor is capable of forming, in the mass of 
the liquid, bubbles which can withstand the superincumbent 
pressure of the liquid and the atmosphere above it. These 
bubbles of vapor, escaping from the liquid, give rise to the 
phenomenon called edudltztion, or boiling. Boiling may, there- 
fore, be defined as the agitation of a liquid by its own vapor. 

Generally speaking, for a given liquid, ebullition always oc- 
curs at the same temperature for the same pressure; and, when 
once commenced, the temperature of the liquid no longer 
rises, no matter how intense the source of heat. This fixed 
temperature is called the dozlzng-point of the liquid. It differs 
for different liquids, and for the same liquid under different 
pressures. That the boiling-point must depend upon the pres- 
sure is evident from the explanation of the phenomenon of 
ebullition above given. 

Substances in solution, if less volatile than the liquid, retard 
ebullition. While pure water boils at 100°, water saturated 
with common salt boils at 109°. The material of the contain- 
ing vessel also influences the boiling point. In a glass vessel 
the temperature of boiling water is higher than in one of metal. 
If water be deprived of air by long boiling, and then cooled, its 
temperature may afterwards be raised considerably above the 
boiling- point before ebullition commences. Under these con- 
ditions, the first bubbles of vapor will form with explosive vio- 
lence. The air dissolved in water separates from it at a high 
temperature in minute bubbles. Into these the water evapo- 
rates, and, whenever the elastic force of the vapor is sufficient 
to overcome the superincumbent pressure, it enlarges them, 
and causes the commotion that marks the phenomenon of 


152] EFFECTS OF HEAT. 183 


ebullition. If no such openings in the mass of the fluid exist, 
the cohesion of the fluid, or its adhesion to the vessel, as well 
as the pressure, must be overcome bythevapor. This explains 
the higher temperature at which ebullition commences when 
the liquid has been deprived of air. 

151. Spheroidal State.—If a liquid Be introduced into a 
highly heated capsule, or poured upon a very hot plate, it does 
not wet the heated surface, but forms a flattened spheroid, 
which presents no appearance of boiling, and evaporates only 
very slowly. Boutigny has carefully studied these phenomena, 
and made known the following facts. The temperature of the 
spheroid is below the boiling-point of the liquid. The spheroid 
does not touch the heated alte but is separated from it bya 
non-conducting layer of vapor. This accounts for the slowness 
of the evaporation. To maintain the liquid in this condition 
the temperature of the capsule must be much above the boil- 
ing-point of the liquid; for water it must be at least 200° C. 
If the capsule be allowed to cool, the temperature will soon fall 
below the limit necessary to maintain the spheroidal state, the 
liquid will moisten the capsule, and there will be a rapid ebul- 
lition, with disengagement of vapor. If a liquid of very low 
boiling-point, as liquid nitrous oxide, which boils at — 88°, be 
poured into,a red-hot capsule, it will assume the spheroidal 
state; and, since its temperature cannot rise above its boiling- 
point, water, or even mercury, plunged into it, will be frozen. 

152. Production of Vapor in a Limited Space.—When a 
liquid is heated in a limited space the vapor generated accumu- 
lates, increasing the pressure, and the temperature rises above 
the ordinary boiling-point. Cagniard-Latour experimented 
upon liquids in spaces but little larger than their own volumes. 
He found that, at a certain temperature, the liquid suddenly 
disappeared; that is, it was converted into vapor in a space 
but little larger than its own volume. It is supposed that 
above the temperature at which this occurs, which is called the 


184 ELEMENTARY PHYSICS. [153 


critical temperature, the substance cannot exist in the liquid 
state. 

153. Liquefaction.—Only a certain amount of vapor can 
exist at a given temperature in a given space. If the tempera- 
ture of a space saturated with vapor be lowered, some of the 
vapor must condense into the liquid state. It is not necessary 
that the temperature of the whole space be lowered ; for, when 
the vapor in the cooled portion is condensed, its pressure is 
diminished, the vapor from the warmer portion flows in, to be 
in its turn condensed, and this continues until the whole is 
brought to the density and pressure due to the cooled portion. 
Any diminution of the space occupied by a saturated vapor 
at constant temperature, will cause some of the vapor to be- 
come liquid, for, if it do not condense, its pressure must in- 
crease; but a saturated vapor is already at its maximum pres- 
sure. 

If the vapor ina given space be not at its maximum pres- 
sure, its pressure will increase when its volume is diminished, 
until the maximum pressure is reached; when, if the tempera- 
ture remain constant, further reduction of volume causes 
condensation into the liquid state, without further increase of 
pressure or density. This statement is true of several of the 
gases at ordinary temperatures. Chlorine, sulphur dioxide, 
ammonia, nitrous oxide, carbon dioxide, and several other 
gases, become liquid under sufficient PPICoate Andrews 
found that, at a temperature of 30.92°, pressure ceases to 
liquefy carbon dioxide. This is the critical temperature for 
that substance. The critical temperatures of oxygen, hydrogen, 
and the other so-called permanent gases, are so low that it is 
only by methods capable of yielding an extremely low tempera- 
ture that they can be liquefied. By the use of such methods 
any of the gases may be made to assume the liquid state. In 
the case of hydrogen, however, the low temperature necessary 
for its liquefaction has only been reached by allowing the gas. 


154] EFFECTS OF HEAT. 185 


to expand suddenly from a condition of great condensation, in 
which it had already been cooled to a very low point. | 

154. Pressure and Density of Non-saturated Gases and 
Vapors.—lf a gas or vapor in the non-saturated condition be 
maintained at constant temperature, it follows very nearly 
Boyle’s law ($$ 77 and 98). If its temperature be below its 
critical temperature, the product of volume by pressure dimin- 
ishes, and near the point of saturation the departure from the 
law may be considerable. At this point there is a sudden 
diminution of volume, and the gas assumes the liquid state. 
The less the pressure and density of the gas, the more nearly 
it obeys Boyle’s law. 

It has been stated already (§ 99) that gases expand as the 
temperature rises. The law of this expansion, called, after its 
discoverer, Gay-Lussac’s law, is that, for each increment of 
temperature of one degree, every gas expands by the same 
constant fraction of its volume at zero. This is equivalent to 
saying that a gas has a constant coefficient of expansion, 
which is the same for all gases. 

Let V,, V;, represent the volumes at zero and ¢ respectively, 
and @ the coefficient of expansion. Then, the pressure remain- 
ing constant, we have 


Via Vit at). (51) 


If @,, d;, represent the densities at the same two tempera- 
tures, we have, since densities are inversely as volumes, 


a, 
1 + at (52) 


poe 


Later investigations, especially those of Regnault, show that 
this simple law, like the law of Boyle, is not rigorously true, 
though it is very nearly so for all gases and vapors which are 


186 ELEMENTARY PHYSICS. [155 


not too near their points of saturation. The common co- 
efficient of expansion is @ = 0.003666 = s4, very nearly. 

Irom the law of Boyle we have, for a given mass of gas, if 
the temperature remain constant, 


V ,p = Vyp’ = volume at pressure unity, 


where V, Vy, represent the volumes at pressure pf and 7’ re- 
spectively. 
From the law of Gay-Lussac we have, if the pressure remain 
constant, 
ZZ Vy 
(Tee ag Sinead es (53) 


If the temperature and pressure both vary, we have 


VIP ES 
ToL ay eee (54) 


that is, if the volume of a given mass of gas be multiplied by 
the corresponding pressure and divided by the factor of ex- 
pansion, the quotient is constant. 

155. Pressure and Density of Saturated Gases and 
Vapors.—It has been seen that, for each gas or vapor at a 
temperature below the critical temperature, there is amaximum 
pressure which it can exert at that temperature. To each 
temperature there corresponds a maximum pressure, which is 
higher as the temperature is higher. A gas or vapor in contact 
with its liquid in a closed space will exert its maximum pressure. 

The relation between the temperature and the correspond- 
ing maximum pressure of a vapor is a very important one, and 
has been the subject of many investigations. The vapor of 
water has been especially studied, the most extensive and 
accurate experiments being those of Regnault. 


155] EFFECTS OF HEAT. 187 


Two distinct methods were employed, one for temperatures 
below 50°, and the other for higher temperatures. The first 
consisted in observing the difference in height of two barom- 
eters placed side by side, the vacuum chamber of one con- 
taining a little water. The temperature was carried from zero 
to about 50°. Both barometers were surrounded by the same 
medium, and in every way under the same conditions, except 
that water and its vapor were present in one and not in the 
other. The difference between the heights of the two gave 
the pressure of the vapor at the temperature of the experiment. 

The second method was founded on the principle that the 
vapor of a boiling liquid exerts a pressure equal to that of the 
atmosphere above it. The experiment consisted in boiling 
water in a closed space in which 
the air could be rarefied or con- 
densed to a known pressure, and 
noting the temperature of the 
boiling liquid and that of the 
vapor above it. To prevent the 
accumulation of the vapor and 
the consequent change of pres- 
sure, a condenser communicated 
with the boiler, consisting of a 
tube surrounded by a larger 
tube, forming an annular space, “O10 20 30 4050 G0 70 80 90 400 
through which a stream of cold Temperatures. 
water was kept flowing. By this yee 
means the vapor was condensed as fast as formed, and the water 
from its condensation flowed back into the boiler. By rarefy:ng 
or compressing the air in the closed space, an artificial atmos- 
phere of any desired pressure could be obtained, and maintained 
constant as long as was necessary for making the observations. 

The temperature was determined by means of four 
thermometers placed in the boiler, two of them in the liquid 


J 
3 
— 
3 
wn 
$ 
x 

& 


188 ELEMENTARY PHYSICS. [156 


and two in the vapor. The bulbs of the thermometers were 
placed in metal tubes, to protect them from the pressure, which 
otherwise would compress the bulb, and cause the thermome- 
ter to register too high a temperature. 

The results of Regnault’s observations may be _ repre- 
sented graphically, as in Fig. 51, where pressures are measured 
in the vertical, and temperatures in the horizontal, direction. 
It is seen that the pressure varies very rapidly with the tem- 
perature. i 

156. Kinetic Theory of Gases.—According to the £znetic 
theory of gases, a perfect gas consists of an assemblage of free, 
perfectly elastic molecules in constant motion. Each mole- 
cule moves in a straight line with a constant velocity, until it 
encounters some other molecule, or the side of the vessel. 
The impacts of the molecules upon the sides of the vessel are 
so numerous that their effect is that of a continuous constant 
force or pressure. 

The entire independence of the molecules is assumed from 
the fact that, when gases or vapors are mixed, the pressure of 
one is added to that of the others; that is, the pressure of the 
mixture is the sum of the pressures of the separate gases. It 
follows from this, that no energy is required to separate the 
molecules; in other words, no internal work need be done to 
expand a gas. This was demonstrated experimentally by 
Joule, who showed that when a gas expands without perform- 
ing external work, it is not cooled. 

The action between two molecules, or between a molecule 
and a solid wall, must be of such a nature that no energy is 
lost; that is, the sum of the kinetic energies of all the mole- 
cules must remain constant. Whatever be the nature of this 
action, it is evident that when a molecule strikes a solid 
stationary wall, it must be reflected back with a velocity equal 
to that before impact. If the velocity be resolved into two 
components, one parallel to the wall and the other normal to 


156] EFFECTS OF HEAT. 1So 


it, the parallel component remains unchanged, while the nor- 
mal component is changed from + 7, its value before impact, 
to — v, its value after impact. The change of velocity is there- 
fore 2v; and if @ represent the duration of impact, the mean 


Be 


~ 


; . 2U : 
acceleration is @ and the mean force. of impact p = m- 


Ae 
where # represents the mass of the molecule. 

Since the effect of the impacts is a continuous pressure, 
the total pressure P exerted upon unit area is equal to this 
mean force of impact of one molecule multiplied by the num- 
ber of molecules meeting unit area in the time 6. To find this 
latter factor, we suppose the molecules confined between two. 
parallel walls at a distance s from each other. Any molecule 
may be supposed to suffer reflection from one wall, pass across 
to the other, be reflected back to the first, and so on. What- 
ever may be the effect of the mutual collisions of the mole- 
cules, the number of impacts upon the surface considered will 
be the same as though each one preserved its rectilinear mo- 
tion unchanged, except when reflected from the solid walls. 
The time required for a molecule moving with a velocity uv to 


ae 
pass across the space between the two walls and back is pee 


and the number of impacts upon the first surface in unit time 
ee 
1S a 

Represent by z’ the number of molecules in a rectangular 
prism, with bases of unit area in the walls. These molecules 
must be considered as moving in all directions and with various 
velocities. But the velocity of any molecule may be resolved 
in the direction of three rectangular axes, one normal to the 
surface and the other two parallel to it; and, since the number 
of molecules in any finite volume of gas is practically infinite, 
the effect upon the wall due to their real motions will be the 
same as would result from a motion of one third the total 


190 : ELEMENTARY PHYSICS. [156 


number of molecules in each of the three directions with the 
mean velocity. Hence the number of molecules moving, in a 
manner similar to that of the single molecule already consid- 
ered, normal to the walls is $2’. The number of impacts upon 


/ 


3 : Pbehee Mes Wee gr . ; 
unit area of the first surface in unit time is 5 Se and in time @ 


8 aa dae : 
is Ae Hence the total pressure P on unit area is 
29) 1 nud I ee 
Weis aor a is Bon oa (Oe 
0 ZirS 3 S 


/ 


But > is the number of molecules in unit volume. Repre- 


senting this by z, we have 
Pan ‘C53 


That is, the pressure upon unit area is equal to one third the 
number of molecules in unit volume at that pressure multiplied 
by twice the kinetic energy of each molecules 

Suppose, now, the volume of the gas be changed from unity 
to V, without change of temperature. The number of mole- 
cules in unit volume is now i and the pressure P, = pm 
whence P,V = jumv*. This is a constant quantity, since z and 
m are constant for the same mass of gas, and v is constant if 
there be no change of temperature. But PV equal to a con- 
stant is Boyle’s law. 

From the law of Gay-Lussac we have, if P represent the 
pressure at 7°, and P. the pressure at zero, 


1 a AC Reena 


156] BPFECT SOF HEAT, Igt 


We have a = sh, very nearly; hence 
If ¢= — 273°, 


that is, at 273° below zero the pressure vanishes. Since 
P = 4tnmv’, it follows that, at this temperature, v = 0, or the 
molecules are at rest. This temperature is therefore called the 
absolute sero. 

In studying the expansion of gases, it is very convenient to 
use a scale of temperatures the zero-point of which is at the 
absolute zero. Temperatures reckoned upon this scale are 
called absolute temperatures. Let Z represent a temperature 
upon the absolute scale: then = 7¢-++ 273, and Eq. (56) be- 


fe 
comes P= Piary Substituting the value of P from (55), we 


have 
4umv" = pies 
273 
whence 
Wr fe WN 
I hm DM MY : (57) 


0 


That is, the absolute temperature of a gas is proportional to 
the kinetic energy of the molecules. 

It has been already stated ($ 100), that, when a gas is com- 
pressed, a certain amount of heat is generated. Suppose a 
cylinder with a tightly-fitting piston. So long as the piston is 


192 ELEMENTARY PHYSICS. [157 


at rest, each molecule that strikes it is reflected with a velocity 
equal to that before impact: but if the piston be forced into 
the cylinder, each molecule, as it is reflected, has its velocity 
increased; and, as was shown above, this is equivalent to a rise 
in temperature. It can be shown that the increase of kinetic 
energy in this case is precisely equal to the work done in forc- 
ing the piston into the cylinder against the pressure of the gas. 
On the other hand, if the piston be pushed backward by the 
force of the impact of the molecules, there will be a loss of 
velocity by reflection from the moving surface, kinetic energy 
equal in amount to the work done upon the piston disappears, 
and the temperature falls. 

The phenomena exhibited by the radzometer afford a strong 
experimental confirmation of the kinetic theory of gases. 
These phenomena were discovered by Crookes. In the form 
first given to it by him, the instrument consists of a delicate 
torsion balance suspended in a vessel from which the air is very 
completely exhausted. On one end of the arm of the torsion 
balance is fixed a light vane, one face of which is blackened. 
When a beam of light falls on the vane, it moves as if a press- 
ure were applied to its blackened surface. The explanation of 
this movement is, that the molecules of air remaining in the 
vessel are more heated when they come in contact with the 
blackened face of the vane than when they come in contact 
with the other face, and are hence thrown off with a greater 
velocity, and react more strongly upon the blackened face of 
the vane. At ordinary pressures the free paths of the mole- 
cules are very small, their collisions very frequent, and any in- 
equality in the pressures is so speedily reduced, that no effect 
upon the vane is apparent. At the high exhaustions at which 
the movement of the vane becomes evident, the collisions are 
less frequent, and hence an immediate equalization of pressure 
does not occur. The vane therefore moves in consequence of 
the greater reaction upon its blackened surface. 


158] EFFECTS OF HEAT. 193 


—_—<—$_ 


157. Mean Velocity of Molecules.— Equation (55) enables 
us to determine the mean velocity of the molecules of a gas of 
which the density and pressure are known, since mm is the 
mass of the gas in unit volume. 

Solving the equation with reference to v, and substituting 
the known values of the constants for hydrogen, namely, 
P= 1013373 dynes per square centimetre, and zz, or density, 
= 0.00008954 grams per cubic centimetre, we have 184260 cen- 
timetres per second, or a little more than one mile per second, 
as the mean velocity of a molecule of hydrogen. 

158. Elasticity of Gases.—It has been shown (§ 77) that 
the elasticity of a gas, obeying Boyle’s law, is numerically equal 
to the pressure. ‘ This is the elasticity for constant temperature. 
But, as was seen ($ 156), when a gas is compressed it is 
heated; and heating a gas increases its pressure. Under ordi- 
nary conditions, therefore, the ratio of a small increase of pres- 
sure to the corresponding decrease of unit volume is greater 
than when the temperature is constant. It is important to 
consider the case when all the heat generated by the compres- 
sion is retained by the gas. ‘The elasticity is then a maximum, 
and is called the elasticity when no heat ts allowed to enter or 
escape. 

Let mun (Fig. 52) be a curve representing the relation be- 
tween volume and pressure for con- 
stant temperature, of which the ab- 
scissas represent volumes and the 
ordinates pressures. Such a curve 
is called an zsothermal line. It is 
plain that to each temperature must 
correspond its own isothermal line. 
If, now, we suppose the gas to be 
compressed, and no heat to escape, 
it is plain that if the volume dimin-o Gc 
ish from OC to OG, the pressure will _ Fie. 52. 


become greater than GD; suppose it to be GAZ. Ifa number 
13 


194 ELEMENTARY PHYSICS. [159 


of such points as 47 be found, and a line be drawn through 
them, it will represent the relation between volume and pres- 
sure when no heat enters or escapes. It is called an adiabatic 
line. It evidently makes a greater angle with the horizontal 
than the isothermal. 

159. Specific Heats of Gases.—lIn § 156 it is seen that the 
temperature of a gas is proportional to the kinetic energy of 
its molecules. To warm a gas without change of volume is, 
therefore, only to add to this kinetic energy. If, however, the 
gas be allowed to expand when heated, the molecules lose 
energy by impact upon the receding surface; and this, together 
with the kinetic energy due to the rise in temperature, must be 
supplied from the source of heat. It has been seen that the 
loss of energy resulting from impact upon a receding surface is 
equal to the work done by the gas in expanding. 

The amount of heat necessary to raise the temperature of 
unit mass of a gas one degree, while the volume remains un- 
changed, is called the specific heat of the gas at constant volume. 
The amount of heat necessary to raise the temperature of unit 
mass of a gas one degree when expansion takes place without 
change of pressure, is called the specific heat of the gas at con- 
stant pressure. 

From what has been said above, it is evident that the differ- 
ence between these two quantities of heat is the equivalent of 
the work done by the expanding gas. 

The determination of the relation of these two quantities is 
a very important problem. 

The specific heat of a gas at constant pressure may be found 
by passing a current of warmed gas through a tube coiled ina 
calorimeter. This is the method of mixtures (§ 111). There 
are great difficulties in the way of an accurate determination, 
because of the small density of the gas, and the time required 
to pass enough of it through the calorimeter to obtain a reason- 
able rise of temperature. The various sources of error produce 


159] EFFECTS OF HEAT. 195 


effects which are sometimes as great as, or even greater than, 
the quantity to be measured. It is beyond the scope of this 
work to describe in detail the means by which the effects of 
the disturbing causes have been determined or eliminated. 

The specific heat of a gas at constant volume is generally 
determined from the ratio between it and the specific heat at 
constant pressure. The first determination of this ratio was 
accomplished by Clement and Desormes. 

The theory of the experiment may be understood from the 
following considerations : 

Let a unit mass of gas at any temperature ¢ and volume 
V, be confined in a cylinder by a closely fitting piston of area 
A. Suppose its temperature to be raised one degree, by com- 
munication of heat from some external source, while its volume 
remains unchanged. It absorbs heat, which we will suppose 
measured in mechanical units, and will represent by C, the 
specific heat at constant volume. Now let the gas expand, at 
the constant temperature ¢-+ I, until it returns to its original 
pressure. During this expansion the piston will be forced out 
through a distance @, and an additional quantity of heat will 
be absorbed from the source. Represent by P the mean 
pressure on unit area of the piston exerted by the gas during 
this operation. Then the work done during expansion, which 
is the equivalent of the heat absorbed, is Pdd. Ad represents 
the increase in volume of the gas during this process. The 
same increase in volume would have occurred had the gas been 
allowed to expand at constant pressure, while its temperature 
was rising. But, for arise in temperature of one degree, the 
increase in. volume of any mass of gas is aV,, where V, repre- 


0 
sents the volume at zero. Hence we have dd = aV,, and the 
work done during the expansion is PAd = PuV,. The heat 
absorbed, therefore, in raising the temperature of the gas one 
degree at constant pressure is C, = C, + PaV,. C, represents 


the specific heat of the gas at constant pressure, measured in 


190 ELEMENTARY PHYSICS. [159 


mechanical units. The ratio of the two specific heats is 


C I 
fait GPal,. (58) 


If, in the case considered above, the gas had expanded 
without receiving any heat, the work PaV, would have been 
done at the expense of its own internal energy, and the 
temperature would have fallen. The performance of this work 
is equivalent to abstracting the quantity of heat, PaV,, which 

I ; 
would lower the temperature EG .PaV, degrees, since thewacs 
val 
straction of a quantity C, of heat would lower the temperature 
one degree. Represent this change of temperature by 6. Re- 
RETA PeL ae. that the supposed change of volume was @V,, which 


a 
equals AR Sy ,and that the original volume was V,, it is seen 


a : F 
that the it) of ————— in unit volume would cause a fall in 


I + at 
temperature of 6 degrees. Substituting @ for aba, in Eqs 


(58), we have A — 1+. It is the object of the experiment 


v 
to find 6. The method of Clement and Desormes is as follows: 

A large flask is furnished with a stopcock having a large 
opening, and a very sensitive manometer which shows the 
difference between the pressure in the flask and the pressure 
of the air. The air in the flask is first rarefied antevierm 
assume the temperature of the surrounding atmosphere. Sup- 
pose its pressure now to be H — 4, # representing the height 
of the barometer, and # the difference between the pressure in 
the flask and the pressure of the atmosphere, as shown by the 
manometer. The large stopcock is then suddenly opened for 
a very short time only; the air rushes in, re-establishes the 


159] EFFECTS OF HEAT. 197 


atmospheric pressure, compresses the air originally in the flask, 
and raises its temperature. ‘The volume of the air becomes 
1 — @, where its original volume is taken as unity and ¢ repre- 
sents its reduction ; and, if there were no change of tempera- 


ff —h : 
ture, the pressure would be oN If the temperature in 


Pp 


crease 0’ degrees, and become ¢-+ 6’, the pressure will be 


H—-h_ t+ta(t+@) 
eon Ome las alts (59) 


the atmospheric pressure. 

The flask is now left until the air within it returns to the 
temperature of the atmosphere ¢, when the manometer shows 
a fall of pressure #’, and we have 

fT —h ; 
eTynnR: = td —— fe (60} 


From these two equations we have 


los ein) g — baer’ 
ee rr i a — 2) 


a 
Suppose, now, the change of volume had been arg: 
then the change of temperature would have been 6; and, since 
change of volume is proportional to change of temperature, we 


have 


} - & — 6’: @: 
Airman ric: o U'» 


hence 


198 ELEMENTARY PHYSICS. [160 


or, substituting the values of ¢ and 6’, we have 


h’ ff—k _ h’ 


ey Ny gt he 


Now we have shown that 


Cs 
Cor 1+ 46; 
hence 
bey he AS ee 
7 ahG ae (oD 


160. The Two Specific Heats of a Gas have the Same 
Ratio as the Two Elasticities.—Suppose a gas, of which 
the mass is unity and volume V, to rise in temperature at 
constant pressure from the temperature ¢ to the temperature 
(¢ + 4¢), 4¢ representing a very small increment of tempera- 
ture. The heat consumed will be C,4z, and the increase of 
volume aV,4¢t. Now, if the volume had remained constant, 
the amount of heat required to cause the rise of temperature 
At would have been C,4¢. Hence if the gas be not allowed to 
expand, the amount of heat, C,47, will cause a rise of tempera- 
Cy 
oe 
gas, after first being allowed to expand, be compressed to its 
initial volume. Such a compression would be attended by an 
increase of pressure, which we will call 44. The ratio between 
this and the corresponding change of volume is 


ture ~~ 4¢; and the same rise of temperature will occur if the 


4 
Weel se (62) 


160] LIPECTS OF HEAT, 199 


where £&, is the elasticity under the condition that no heat 
enters or escapes. 

If, now, the heat produced by compression be allowed to 
escape, there will remain the quantity C,47, and the increment 


. Ce . . . 
of pressure will be reduced to 0g = Ape. This is the increase 
2 
of pressure that will occur if the gas be compressed bythe 
amount a)V,4¢ without change of temperature; hence 


“) 
NCR se a Ts (63) 


where 4; is the elasticity for constant temperature. Dividing 
(62) by (63), we have 


Ap 
E, a Al ap. Ap be Gs) 
ia 7 SE eC ROM 
aV.At Lee 


that is, the two elasticities have the same ratio as the two 
specific heats of a gas. 

It may be shown that the velocity of sound in any medium 
is equal to the square root of the quotient of the elasticity 
divided by the density of the medium; that is, 


velocity = / = (64) 


In the progress of a sound-wave, the air is alternately com- 


200 ~* ELEMENTARY PHYSICS. [161 


pressed and rarefied, the compressions and rarefactions occur- 
ring in such rapid succession that there is no time for any 
transfer of heat. If Eq. (64) be applied to air, the & becomes 
£,, or the elasticity under the condition that no heat enters or 
escapes. Since we know the density of the air and the velocity 
of sound, A, can be computed. In§77 it is shown that &, is 
numerically equal to the pressure; hence we have the values 
of the two elasticities of air, and, as seen above, their ratio is 
the ratio of the two specific heats of air. | 

* 161. Examples of Energy absorbed by Vaporization.— 
When a liquid boils, its temperature remains constant, however 
intense the source of heat. This shows that the heat applied 
to it is expended in producing the change of state. Heat is 
absorbed during evaporation. By promoting evaporation, in- 
tense cold may be produced. In a vacuum, water may be 
frozen. by its own evaporation. If a liquid be heated to a 
temperature above its ordinary boiling-point under pressure, 
relief of the pressure is followed by a very rapid evolution of 
vapor and a rapid cooling of the liquid. Liquid nitrous oxide 
at a temperature of zero is still far above its boiling-point, and 
its vapor exerts a pressure of about thirty atmospheres. If the 
liquid be drawn off into an open vessel, it at first boils with 
extreme violence, but is soon cooled to its boiling-point for the 
atmospheric pressure, about — 88°, and then boils away slowly, 
while its temperature remains at that low point. 

162, Heat Equivalent of Vaporization.—It is plain, from 
what has preceded (§ 148), that the formation of vapor is work 
requiring the expenditure of energy for its accomplishment. 
Each molecule that is shot off into space obtains the motion 
which projected it beyond the reach of the molecular attrac- 
tion, at the expense of the energy of the molecules that remain 
behind. A quantity of heat disappears when a liquid evapo- 
. rates; and experiment demonstrates, that to evaporate a kilo- 
gram of a liquid at a given temperature always requires the 


164] BPP EG lo Wl ee iA TL. 201 


same amount of heat. This is the heat eguivalcnt of vaporiza- - 
zion. When a vapor condenses into the liquid state, the same 
amount of heat is generated as disappears when the liquid 
assumes the state of vapor. The heat equivalent of vaporiza- 
tion is determined by passing the vapor at a known tempera- 
ture into a calorimeter, there condensing it into the liquid 
state, and noting the rise of temperature in the calorimeter. 
This, it will be seen, is.essentially the method of mixtures. 
Many experimenters have given attention to this determina- 
tion; but here, again, the best experiments are those of Reg- 
nault. He determined what he called the /otal heat of steam 
at various pressures. By this was meant the heat required to 
raise the temperature of a kilogram of water from zero to the 
temperature of saturated vapor at the pressure chosen, and 
then convert it wholly into steam. The result of his experi- 
ments give, for the heat equivalent of vaporization of water at 
100°, 537 calories. That is, he found that by condensing a 
kilogram of steam at 100° into water, and then cooling the 
water to zero, 637 calories were obtained. But almost exactly 
100 calories are derived from the water cooling from 100° to 
zero; hence 537 calories is the heat equivalent of vaporization 
at 100°. 

163. Dissociation.—It has already been noted (§ gg), that, 
at high temperatures, compounds are separated into their ele- 
ments. To effect this separation, the powerful forces of chem- 
ical affinity must be overcome, and a considerable amount of 
energy must be consumed. 

164. Heat Equivalent of Dissociation and Chemical 
Union.—From the principle of the conservation of energy, it 
may be assumed that the energy required for dissociation is 
the same as that developed by the reunion of the elements. 
The heat equivalent of chemical union is not easy to deter- 
mine because the process is usually complicated by changes 
of physical state. We may cause the union of carbon and 


202 ELEMENTARY PHYSICS. [165 


oxygen in a calorimeter, and, bringing the products of com- 
bustion to the temperature of the elements before the union, 
measure the heat given to the instrument; but the carbon has 
changed its state from a solid to a gas, and some of the chem- 
ical energy must have been consumed in that process. The 
heat measured is the avazlable heat. The best determinations 
of the available heat of chemical union have been made by 
Andrews, Favre and Silbermann, and Berthelot. 


HYGROMETRY. 


165. Object of Hygrometry.—Hygrometry has for its ob- 
ject the determination of the state of the air with regard to 
moisture. 

The amount of vapor in a given volume of air may be de- 
termined directly by passing a known volume of air through 
tubes containing some substance which will absorb the mois- 
ture, and finding the increase in weight of the tubes and their 
contents. The quantity of vapor contained in a cubic metre 
of air is called its absolute humidity. Methods of determining 
this quantity indirectly are given below. 

166. Pressure of the Vapor.—It has been seen (§ 149), 
that, when two or more gases occupy the same space, each 
exerts its own pressure independently of the others. The pres- 
sure of the atmosphere is, therefore, the pressure of the dry air, 
with that of the vapor of water added. If we can determine 
this latter pressure it is easy to compute the quantity of mois- 
ture in the air. 

It has also been seen that the pressure exerted by the 
vapor in the air is at a certain temperature its maximum pres- 
sure. Now, if any small portion of the space be cooled till its 
temperature is below that at which the pressure exerted is the 
maximum pressure, a portion of the vapor will condense into 
liquid. If, then, we determine the temperature at which con- 
densation begins, the maximum pressure of the vapor for this 


167] 'BEnnOl a OF LAT, 203 


temperature, which may be found from tables, is the real pres- 
sure of the vapor inthe air. The mass of vapor ina cubic 
metre of air may then be computed as follows: A cubic metre 
of dry air has a mass of 1293.2 grams at zero and at 700 milli- 
metres pressure. At the pressure # of the vapor, and tem- 
perature ¢ of the air at the time of the experiment, the same 
space would contain 


7203.2 X p : 

93° 760 * 1+ at 
grams of air; and, since the density of vapor of water referred 
to air is 0.623, a cubic metre would contain 


P 


oe SKS 760 ** I a 


X 0.623 (65) 


grams of vapor. 

167. Dew Point.—The temperature at which the vapor of 
the air begins to condense is called the dew point. It is deter- 
mined by means of instruments called dew-point hygrometers, 
which are instruments so constructed that a small surface 
exposed to the air may be cooled until moisture deposits 
upon it, when its temperature is accurately determined. The 
Alluard hygrometer consists of a metal box about one anda 
half centimetres square and four centimetres deep. Two tubes 
pass through the top of the box—one terminating just inside 
and the other extending to the bottom. One side of the box 
is gilded and polished, and is so placed that the gilded surface 
lies on the same plane with, and in close proximity to, a gilded 
metal plate. The box is partly filled with ether, and the short 
tube is connected with an aspirator. Air is thus drawn through 
the longer tube, and, bubbling up through the ether, causes 
rapid evaporation, which soon cools the box, and causes a 
deposit of dew upon the gilded surface. The presence of the 
gilded plate helps very much in recognizing the beginning of © 
the deposit of dew, by the contrast between it and the dew- 


204 ELEMENTARY PAYSICS. [168 


covered surface of the box. A thermometer plunged in the 
ether gives its temperature, and another outside gives the tem- 


perature of the air. The temperature of the ether is the dew , 


point. From it the pressure of the vapor in the air is deter- ‘ 


mined as described in the last section, and this pressure sub- 
stituted for in Eq. 65 gives the absolute humidity. 

168. Relative Humidity.—The amount of moisture that 
the air may contain depends upon its temperature. The damp- 
ness or dryness of the air does not depend upon the absolute 
amount of moisture it’ contains, but upon the ratio of this to 
the amount it might contain if saturated. The relative humtd- 
azty is the ratio of the amount of moisture in the air to that 
which would be required to saturate it at the existing temper- 
ature. Since non-saturated vapors follow Boyle’s law very 
closely, this ratio will be very nearly the ratio of the actual 
pressure to the possible pressure for the temperature. Both 
these pressures may be taken from the tables. One corre- 
sponds to the dew point, and the other to the temperature of 
the air 


« 


a ees 
; 


CHARTER LV: 


THERMODYNAMICS. 


169. First Law of Thermodynamics.—The first law of 
thermodynamics may be thus stated: When heat is trans- 
formed into work, or work into heat, the quantity of work is. 
equivalent to the quantity of heat. The experiments of Joule 
and Rowland establishing this law, and determining the me- 
chanical equivalent, have already been described (§ 114). 

170. Second Law of Thermodynamics.—When heat is 
converted into work by any heat-engine under the conditions 
that exist on the earth’s surface, only a comparatively small 
proportion of the heat drawn from the source can be so trans- 
formed. The remainder is given up to a refrigerator, which in 
some form must be an adjunct of every heat-engine, and still 
exists as heat. It will be shown that.the heat which is con- 
verted into work bears to that which must be drawn from the 
source of heat a certain simple ratio depending upon the tem- 
peratures of the source and refrigerator. The second law of 
thermodynamics asserts this relation. The ratio between the 
heat converted into work and that drawn from the source is 
called the eficzency of the engine. 

To convert heat.into mechanical work, it is necessary that 
the heat should act through some substance called the working 
substance ; as for instance, steam in the steam-engine or air in 
the hot-air engine. In studying the transformation of heat 
into work, it is an essential condition that the working sub- 
stance must, after passing through a cycle of operations, return 
to the same condition as at the beginning; for if the substance 
be not in the same condition at the end as at the beginning, 
internal work may have been done, or internal energy expend-~ 


206 ELEMENTARY PHYSICS. [170 


ed, which will increase or diminish the work apparently de- 
veloped from the heat. 

To develop the second law of thermodynamics, we make 
use of a conception due to Carnot, of an engine completely re- 
versible in all its mechanical and physical operations. In the 
discussion of the reverszble cngine we employ a principle, first 
enunciated by Clausius. Clausius’ principle is, that heat cannot 
pass of itself froma cold toa hot body. In many cases this 
principle agrees with common experience, and in other cases 
results in accordance with it have been obtained by experiment. 
It is so fundamental that it is often called the second law of 
thermodynamics. 

Suppose a heat-engine in operation, running forward. It 
will receive from a source a certain quantity of heat A, transfer 
toa refrigerator a certain quantity of heat £, and perform a cer- 
tain amount W of mechanical work. If it be perfectly reversible, 
it will, by the performance upon it of the amount of work W, 
take from the refrigerator the quantity of heat “, and restore to 
the source the amount /. Such an engine will convert into 
work, under given conditions, as large as possible a proportion of 
the heat taken from the source. For, let there be two engines, 
A and B, of which &Z is reversible, working between the same 
source and refrigerator. If possible let A perform more work 
than B, while taking from the source the same amount of heat. 
If Whbe the work it performs, and w the work B performs, B 
will, from its reversibility, by the performance upon it of the 
work w, less than W, restore to the source the amount of heat, 
Hf, which it takes away when running forward. Let A be em- 
ployed to run & backward: A will take from the source a 
‘quantity of heat, H, and perform work, W. BS will restore 
the heat H to the source by the performance upon it of work, 
zw. The system will then continue running, developing the 
work W—w, while the source loses no heat. It must be, 
then, that A gives up to the refrigerator less heat than B takes 


———————— es, *? 


= 


170] THERMODYNAMICS. 207 


away; and the refrigerator must be growing colder. For the 
purposes of this discussion, we may assume that all surround- 
ing bodies, except the refrigerator, are at the same tempera- 
ture as the source; hence the work W—w, performed by the 
system of two engines, must be performed by means of heat 
taken from a body colder than all surrounding bodies. Now 
this is contrary to the principle of Clausius. The hypothesis 
with which we started must, therefore, be false; and we must 
admit that no engine, no contrivance for converting heat 
into work, can under similar conditions, and while taking the 
same heat from the source, perform more work than a rever- 
sible engine. It follows that all reversible engines, whatever 
the working substance, have the same efficiency. This is a 
most important conclusion. In view of it, we may, in study- 
ing the conversion of heat into work, choose for the working 
substance the one which presents the greatest advantage for 
the study. Since of all substances the properties of gases are 
best known, we will assume a perfect gas as the working sub- 
stance. The cycle of four operations which we will study is 
perfectly reversible. It is known as Carno?’s Cycle. 

Suppose the gas to be enclosed in a cylinder having a 
tightly-fitting piston. Suppose the cycle to begin by a depres- 
sion of the piston, compressing the gas, without loss or gain 
of heat, until the temperature rises from 4 to ¢; where ¢ repre- 
sents the temperature of the source, and @ that of the refriger- 
ator. In Fig. 53, let Oa represent the 
volume, and Aa the pressure at the be- 
ginning.. If the gas be compressed un- 
til its volume becomes Q4@, its pressure 
will be 62. AB representing the pres- | 
sures and corresponding volumes dur- ! : 
ing the operation, is an adiabatic line. 6—-;—,--3 7G 
This is the first operation. For the Fic. 53. 
second operation, \et the piston rise, and the volume increase 


ne D 


208 ELEMENTARY PHYSICS. [170 


from 6 to ¢ at the constant temperature of the source. The 
pressure will fall from 08 to cC. SC is the isothermal line for 
the temperature ¢. During this operation, a quantity of heat 
represented by A must be taken from the source, to maintain 
the constant temperature ¢. For the third operation let the 
piston still ascend, and the volume increase from Oc to Od with- 
out loss or gain of heat until the temperature falls from ¢ to 6, 
the temperature at which the cycle began. CVD is an adiabatic 
line. For the fourth operation, let the piston be depressed to 
the starting point, and the gas maintained at the constant 
temperature 4 of the refrigerator. The volume becomes Oa 
and the pressure @d, as at the beginning. DA is the isother- 
mal line for the temperature 6. : 

Now let us consider the work done in each operation. While 
the piston is being depressed through the volume represented 
by ad, work must be performed upon it equal to ad X the 
mean pressure exerted upon the piston. This mean pressure 
lies between Aa and Sd, and the product of this by aé is evi- 
dently the area Ada. In the same way it is shown that 
when the gas expands from @ to ¢ it performs work represented 
by the area BCcb; and again, in the third operation, it performs 
work represented by CDac. In the fourth operation, when the 
gas is compressed, work must be done upon it represented by 
the area dDda. During the cycle, therefore, work is done by 
the gas represented by the area GC Ddé, and work is done upon 
the gas represented by the area LADdb. The difference. rep- 
resented by the area 4BCD is the work done by the engine 
during the cycle. Since the gas is in all respects in the same 
condition at the end as at the beginning of the cycle, no work 
can have been developed from it; and the work which the en- 
gine has done must have been derived from the heat communt- 
cated to the gas during the second operation. 

Now it has been shown that when a gas expands no inter- 
nal work is done in separating the molecules, and when it ex- 


170] THERMODYNAMICS. 2090 


pands at constant temperature no change occurs in the in- 
ternal kinetic energy; the heat which is imparted to the gas 
during the second operation is, therefore, the equivalent to the 
work done by the gas upon the piston, and may be represented 
by the area BCcb. It will be seen, also, that the heat given up 
to the refrigerator during the fourth operation is represented 
by the area ADda, and that heat, the equivalent of the work 
performed by the engine, represented by the area ABCD, has 
disappeared. Of the heat withdrawn from the source, then, 
area ABCD 
area BCch 
fraction is the efficiency of the engine. 

Now let the operation of the cycle be reversed. Starting 
with the volume Oa the gas expands at the temperature 6, ab- 
sorbs a quantity of heat represented by /, the same as it gave 
up when compressed, and performs work represented by 4 Daa; 
next, it is compressed, without loss of heat, until its tempera- 
ture rises to ¢, and work represented by DCcd is done upon 
it; next, it is still further compressed at the temperature /¢, 
until its volume becomes Q@, and its pressure Bb. During this 
operation it gives up the heat H which it absorbed during the 
direct action, and work represented by CBéc is done upon it. 
Lastly, it expands to the starting-point, and falls to its initial 
temperature. It will be seen that each operation is the reverse 
in all respects of the corresponding operation of the direct 
action, and that during the cycle work represented by the area 
ABCD must be performed upon the engine while the quantity 
of heat # is taken from the refrigerator, and the quantity of 
heat / is transferred to the source. Such an engine is therefore 
a reversible engine; and it converts into work as large a pro- 
portion of the heat derived from the source as is possible under 
the circumstances. An inspection of the figure shows that, 
since the line BC remains the same so long as the amount of 
heat H and the temperature ¢ of the source remain constant, | 

14 


only the. fraction is converted into work. This 


210 ELEMENTARY PYfXOSLCS. [=70 


the only way to increase the proportion of work derived from 
a given amount of heat / is to increase the difference of 
temperature between the source and the refrigerator; that is, 
to increase the area 4 BCD, the line AD must be taken lower 
down. The proportion of heat which can be converted into 
work depends, therefore, upon the difference of temperature 
between source and refrigerator. “To determine the nature of 
this dependence, suppose the range of temperature so small 
that the sides of the figure d4CD may be considered straight 
_ and parallel. Produce AV to e, and draw gh representing the 
mean pressure for the second operation. Now ABCD = eBCf 
= Bex bec=gix be. Also BCcb= gh X bc. Thennwertiae 


Hl—h__ar¢a ABCD 2? Ge 


H” area Bre. chee 


But gh is the pressure corresponding to volume Of and tem- 
perature ¢, #z is the pressure corresponding to the same volume 
and temperature 6. These pressures are proportional to the 
absolute temperatures ($ 156); that is, if ¢and @ are tempera- 
tures on the absolute scale, 


ih 0 

care 
and 

Bs dh Pe SUN yi fet! 

PeeaGy Maar (66) 
hence 


k area ABCD t—8@ 
HELO ae area BUch’ ~ (ohms (67) 


In another form the result contained in Eq. (66) may be 


written 


FORMAT ta 


170] LHERMODYNAMICS. 251 


This proportion has been derived upon the supposition that 
the range of temperature was very small: but it is equally true 
for any range; for, let there be a series of engines of small 
range, of which the second has for a source the refrigerator of 
the first, the third has for a source the refrigerator of the 
second, and soon. ‘The first takes from the source the heat 
ff, and gives to the refrigerator the heat 4, working between 
the temperatures ¢ and 6 The second takes the heat # from 
the refrigerator of the first, and gives to its own refrigerator 
the heat “,, working between the temperatures 6 and @,.. The 
operation of the others is similar; then, from Eq. 68, we have 


hs 
Jah Ogee 
AG 
AG? 
r, 8, 
Heh 0; 
h,, — Gn 
ie SE eee 
multiplying, we obtain 
weit, 1 ea ra a: 6, 
TEST SES OY MOA Wee any EE 
or 
Eat 
FT othe 
and 


Hence it appears that, in a perfect heat-engine, the heat con- 


212 ELEMENTARY Yael, [I7Xr 


verted into work is to the heat received as the difference of 
temperature between the source and the refrigerator is to the 
absolute temperature of the source. This ratio can become 
unity only when 6=0°, or when the refrigerator is at the ab- 
solute zero of temperature. Since the difference of tempera- 
tures between which it is practicable to work is always small 
compared to the absolute temperature of the source, a perfect 
heat-engine can convert into work only a small fraction of the 
heat it receives. 

The formulas developed in this section embody what we 
have called the second law of thermodynamics. 

171. Absolute Scale of Temperatures.—An absolute scale 
of temperatures, formed upon the assumed properties of a 
perfect gas, has already been described (§ 156). No such sub- 


Fic. 54. 


stance as a perfect gas exists; but, since (§ 170) any two 
temperatures on the absolute scale are to each other as the 
heat taken from the source is to the heat transferred to the 
refrigerator by a reversible engine, any substance of which we 
know the properties with sufficient exactness to draw its iso- 
thermal and adiabatic lines, may be used as a thermometric 


71] THERMODYNAMICS. 213 


substance, and, by means of it, an absolute scale of tempera- 
tures may be constructed. For example, in Fig. 54 let BA’ be 
an isothermal line for some substance, corresponding to the 
temperature ¢ of boiling water at astandard pressure. Let 6/’ 
‘be the isothermal line for the temperature 7, of melting ice, 
and let 40’ be an isothermal line for an intermediate tempera- 
ture. Let Bf, B’f’, be adiabatic lines, such that, if the sub- 
stance expand at constant temperature ¢ from the condition 
B to the condition 5’, the equivalent in heat of one mechani- 
cal unit of energy will be absorbed. Now, the figure BB’ A's 
‘represents Carnot’s cycle, and the heat given to the refrigerator 
at the temperature 7,, measured in mechanical units, is less 
than the heat taken from the source at the temperature 7, by 
the energy represented by the area B4’f’'Z; or, the heat given 
to the refrigerator is equal to 1 — area Bf’: hence 


1 — arca Bf’ 
I 


Z, 
Z 


3 


and 


¢—7, area Bf 
ie HL 


Now, if ¢ — ¢, = 100°, as in the Centigrade scale, we have 
100 area Bf’ 
Ley Roe 


and 


ia area BB” (69) 


214 ELEMENTARY PHYSICS. (r72 


If 6 be the temperature corresponding to the isothermal 
line 40’, we have, as above, | 


1 — area Bb’ 
Thy 9 


I 


6 
Z 
whence 


6 = ¢(1 — area BO’) = ae — area Bb’). (70) 


area BBS 


If, now, it be proposed to use the substance as a thermometric 
substance by noting its expansion at constant pressure, take 
Om to represent that pressure, and draw the horizontal line 
mnop; mn is the volume of the substance at temperature /7,, 
mo the volume at temperature 6, and mp the volume at tem-— 
perature 7. 

This method of constructing an absolute scale of tempera~ 
ture was proposed by Thomson. 

172. The Steam-Engine.—The steam-engine in its usual 
form consists essentially of a piston, moving in a closed cylin- 
der, which is provided with passages and valves by which steam 
can be admitted and allowed to escape. A boiler heated by a 
suitable furnace supplies the steam. The valves of the cylin- 
der are opened and closed automatically, admitting and dis- 
charging the steam at the proper times to impart to the piston 
a reciprocating motion, which may be converted into a circular 
motion by means of suitable mechanism. 

There are two classes of steam-engines, condensing and 
non-condensing. In condensing engines the steam, after doing 
its work in the cylinder, escapes into a condenser, kept cold by 
a circulation of cold water. Here the steam is condensed into 
water; and this water, with air or other contents of the con- 
- denser, is removed by an “air-pump.’’ In non-condensing 
engines the steam escapes into the open air. In this case the 


ee Po nate a 


173] THERMODYNAMICS. 215 


temperature of the refrigerator must be considered at least as 
high as that of saturated steam at the atmospheric pressure, or 
about 100°, and the temperature of the source must be taken 
as that of saturated steam at the boiler pressure. Applying 
the expression for the efficiency ($170), 


it will be seen. that, for any boiler pressure which it is safe to 
employ in practice, it is not possible, even with a perfect en- 
gine, to convert into work more than about fifteen per cent of 
the heat used. 

In the condensing engine the temperature of the refrigera- 
tor may be taken as that of saturated steam at the pressure 
which exists in the condenser, which is usually about 30° or 
40°: hence ¢ — @ is a much larger quantity for condensing than 
for non-condensing engines. The gain of efficiency is not, 
however, so great as would appear from the formula, because 
of the energy that must be expended to maintain the vacuum 
in the condenser. 

173. Hot-air and Gas Engines.—Hot-air engines consist 
essentially of two cylinders of different capacities, with some 
arrangement for heating air in, or on its way to, the larger 
cylinder. In one form of the engine, an air-tight furnace forms 
the passage between the two cylinders, of which the smaller 
may be considered as a supply-pump for taking air from out- 
side, and forcing it through the furnace into the larger cylinder, 
where, in consequence of its expansion by the heat, it is enabled 
to perform work. On the return stroke, this air is expelled 
into the external air, still hot, but at a lower temperature than 
it would have been had it not expanded and performed work. 
This case is exactly analogous to that of the steam-engine, in 
which water is forced by a piston working in a small cylinder, 


216 ELEMENTARY: PHYSICS. 1173 


into a boiler, is there converted into steam, and then, acting 
upon a much larger piston, performs work, and is rejected. In 
another form of the engine, known as the ‘‘ready motor,” the 
air is forced into the large cylinder through a passage kept sup- 
plied with crude petroleum. The air becomes saturated with 
the vapor, forming a combustible mixture, which is burned in 
the cylinder itself. 

The Stirling hot-air engine and the Rider “ compression 
engine” are interesting as realizing an approach to Carnot’s 
cycle. 

These engines, like those described above, consist of two 
cylinders of different capacities, in which work air-tight pistons; 
but, unlike those, there are no valves communicating with the 
external atmosphere. Air is not taken in and rejected; but 
the same mass of air is alternately heated and cooled, alter- 
nately expands and contracts, moving the piston, and _ per- 
forming work at the expense of a portion of the heat imparted 
to it. 

It is of interest to study alittle more in detail the cycle of 
operations in these two forms of engines. The larger of the 
two cylinders is kept constantly at a high temperature by 
means of a furnace, while the smaller is kept cold by the circu- 
jation of water. The cylinders communicate freely with each 
other. The pistons are connected to cranks set on an axis, so 
as to make an angle of nearly ninety degrees with each other. 
Thus both pistons are moving for a short time in the same © 
direction twice during the revolution of the axis. At the in- 
stant that the small piston reaches the top of its stroke, the 
large piston will be near the bottom of the cylinder, and de- 
scending. The small piston now descends, as well as the 
large one, the air in both cylinders is compressed, and there is 
but little transfer from one to the other. There is, therefore, 
comparatively little heat given up. The large piston, reaching 
its lowest point, begins to ascend, while the descent of the 


174] THERMODYNAMICS. 217 


smaller continues. The air is rapidly transferred to the larger 
heated cylinder, and expands while taking heat from the highly 
heated surface. After the small piston has reached its lowest 
point, there is a short time during which both the pistons are 
rising and the air expanding, with but little transfer from one 
cylinder to the other, and with a relatively small absorption of 
heat. When the descent of the large piston begins, the small 
one still rising, the air is rapidly transferred to the smaller 
cylinder: its volume is diminished, and its heat is given up to 
the cold surface with which it is brought in contact. The 
completion of this operation brings the air back to the condi- 
tion from which it started. It will be seen that there are here 
four operations, which, while not presenting the simplicity of 
the four operations of Carnot,—since the first and third are 
not performed without transfer of heat, and the second and 
fourth not without change of temperature,—still furnish an 
example of work done by heat through a series of changes in 
the working substance, which brings it back, at the end of each 
revolution, to the same condition as at the beginning. 

Gas-engines derive their power from the force developed by 
the combustion, within the cylinder, of a mixture of illuminat- 
ing gas and air. 

As compared with steam-engines, hot-air and gas engines 
use the working substance at a much higher temperature. 
z — @ is, therefore, greater, and the theoretical efficiency higher. 
There are, however, practical difficulties connected with the 
lubrication of the sliding surfaces at such high temperatures 
that have so far prevented the use of large engines of this 
class. 

174. Sources of Terrestrial Energy.— Water flowing from 
a higher toa lower level furnishes energy for driving machin- 
ery. The energy theoretically available in a given time is the 
weight of the water that flows during that time multiplied by 
the height of the fall. If this energy be not utilized, it devel- 


218 ELEMENTARY PHYSICS. [174 


ops heat by friction of the water or of the material that may 
be transported by it. But water-power is only possible so long 
as the supply of water continues. The supply of water is de- 
pendent upon the rains; the rains depend upon evaporation; 
and evaporation is maintained by solar heat. The energy of 
qwater-power is, therefore, transformed solar energy. 

A moving mass of air possesses energy equal to the mass 
multiplied by half the square of the velocity. This energy is 
available for propelling ships, for turning windmills, and for 
other work. Winds are due to a disturbance of atmospheric 
equilibrium by solar heat; and the energy of wénudpower, 
like that of water-power, is, therefore, derived from solar 
energy. | 

The occan currents also possess energy due to their motion, 
and this motion is, like that of the winds, derived from solar 
energy. | 

By far the larger part of the energy employed by man for 
his purposes is derived from the combustion of wood and coal. 
This energy exists as the potential exergy of chemical separation 
of oxygen from carbon and hydrogen. Now, we know that 
vegetable matter is formed by the action of the solar rays 
through the mechanism of the leaf, and that coal is the carbon 
of plants that grew and decayed in a past geologicalage. The 
energy of wood and coal is, therefore, the transformed energy 
of solar radiations. 

It is well known that, in the animal tissues, a chemical 
action takes place similar to that involved in combustion. The 
oxygen taken into the lungs and absorbed by the blood com- 
bines by processes with which we are not here concerned with 
the constituents of the food. Among the products of this 
combination are carbon dioxide and water, as in the combus- 
tion of the same substances elsewhere. Lavoisier assumed that 
such chemical combinations were the source of anzzmal heat, 
and was the first to attempt a measurement of it. He com- 


174] THERMODYNAMICS. 219 


pared the heat developed with that due to the formation of 
the carbonic dioxide exhaled in’a given time.’ Despretz and 
Dulong made similar experiments with more perfect apparatus, 
and found that the heat produced by the animal was about 
one tenth greater than would have been produced by the 
formation by combustion of the carbonic acid and water ex- 
haled. | 

These and similar experiments, although not taking into 
account all the chemical actions taking place in the body, leave 
no doubt that animal heat is due to atomic and molecular: 
changes within the body. 

The work performed by muscular action is also the trans- 
formed energy of food. Rumford, in 1798, saw this clearly; 
and he showed, ina paper’ of that date, that the amount of 
work done by a horse is much greater than would be obtained 
by using its food as fuel for a steam-engine. 

Mayer, in 1845, held that an animal is a heat-engine, that 
every motion of the animal isa transformation into work of the 
heat developed in the tissues. 

Hirn, in 1858, executed a series of interesting experiments 
bearing upon this subject. Ina closed box was placed a sort 
of treadmill, which a man could cause to revolve by stepping 
from step to step. He thus performed work which could be 
measured by suitable apparatus outside the box. The tread- 
wheel could also be made to revolve backward by a motor 
placed outside, when the man descended from step to step, and 
work was performed upon him. 

Three distinct experiments were performed; and the amount 
of oxygen consumed by respiration, and the heat developed, 
were determined. 

In the first experiment the man remained in repose; in the 
second he performed work by causing the wheel to revolve ; in 
the third the wheel was made to revolve backward, and work 
was performed upon him. In the second experiment, the 


220 BLEMEN TARY PRY SICS. [174 


amount of heat developed for a gram of oxygen consumed *was 
much less, and in the third case much greater, than in the first; 
that is, in the first case, the heat developed was due to a chemi- 
cal action, indicated by the absorption of oxygen; in the second, 
a portion of the chemical action went to perform the work, and 
hence a less amount of heat was developed; while in the third 
case the motor, causing the treadwheel to revolve, performed 
work, which produced heat in addition to that due to the 
chemical action. 

It has been thought that muscular energy is due to the 
waste of the muscles themselves: but experiments show that 
the waste of nitrogenized material is far too small in amount 
to account for the energy developed by the animal: and we 
must, therefore, conclude that the principal source of muscular 
energy is the oxidation of the non-nitrogenized material of the 
blood by the oxygen absorbed in respiration. 

An animal is, then, a machine for converting the potential 
energy of food into mechanical work: but he is not, as Mayer 
supposed, a heat-engine; for he performs far more work than 
could be performed by a perfect heat-engine, working between 
the same limits of temperature, and using the food as fuel. 

The food of animals is of vegetable origin, and owes its 
energy to the solar rays. Animal heat and energy is, therefore, 
the transformed energy of the sun. 

The ¢zdes are mainly caused by the attraction of the moon 
upon the waters of the earth. If the earth did not revolve 
upon its axis, or, rather, if it always presented one face to the 
moon, the elevated waters would remain stationary upon its 
surface, and furnish no source of energy. But as the earth 
revolves, the crest of the tidal wave moves apparently in the 
opposite direction, meets the shores of the continents, and 
forces the water up the bays and rivers, where energy is wasted 
in friction upon the shores or may be made use of for turning 
mill-wheels. It is evident that all the energy derived from the 


176] THERMODVNAMICS. 22r 


tides comes from the rotation of the earth upon its axis; and 
a part of the energy of the earth’s rotation is, therefore, being 
dissipated in the heat of friction they cause. 

The internal heat of the earth and a few other forms of 
energy, such as that of native sulphur, iron, etc., are of little 
consequence as sources of useful energy. They may be con- 
sidered as the remnants of the original energy of the earth. 

175. Energy of the Sun.—It has been seen that the sun’s: 
rays are the source of all the forms of energy practically avail- 
able, except that of the tides. It has been estimated that the 
heat received by the earth from the sun each year would melt 
a layer of ice over the entire globe a hundred feet in thickness. 
This represents energy equal to one horse-power for each fifty 
square feet of surface, and the heat which reaches the earth is 
only one twenty-two-hundred-millionth of the heat that leaves 
the sun. Notwithstanding this «enormous expenditure of en- 
ergy, Helmholtz and Thomson have shown that the nebular 
hypothesis, which supposes the solar system to have originally 
existed as a chaotic mass of widely separated gravitating par- 
ticles, presents to us an adequate source for all the energy of 
the system. As the particles of the system rush together by 
their mutual attractions, heat is generated by their collision; 
and after they have collected into large masses, the conden- 
sation of these masses continues to ¢>nerate heat. 

176. Dissipation of Energy.—It has been seen that only 
a fraction of the energy of heat is available for transformation 
into other forms of energy, and that such transformation is. 
possible only when a difference of temperature exists. Every 
conversion of other forms of energy into heat puts it in a form 
from which it can be only partially recovered. Every transfer 
of heat from one body to another, or from one part to another 
of the same body, tends to equalize temperatures, and to 
diminish the proportion of energy available for transformation. 
Such transfers of heat are continually taking place; and, so 


pe it 2 ae ELEMENTARY LA YSIS. [176 


far as our present knowledge goes, there is a tendency toward 
an equality of temperature, or, in other words, a uniform mo- 
lecular motion, throughout the universe. If this condition of 
things were reached, although the total amount of energy 
existing in the universe would remain unchanged, the possibil- 
ity of transformation would be at an end, and all activity and 
change would cease. ‘This is the doctrine of the dissipation of 
energy to which our limited knowledge of the operations of 
nature leads us; but it must be remembered that our knowl- 
edge is very limited, and that there may be in nature the 
means of restoring the differences upon which all activity de- 
pends. 


MAGNETISM AND ELECTRICITY. 


Eg eg Ana 6d kaa 
MAGNETISM. 


177. Fundamental Facts.—Masses of iron ore are some- 
times found which possess the property of attracting pieces of 
iron and a few other substances. Such masses are called wa/u- 
ral magnets or lodestones. A bar of steel may be so treated 
as to acquire similar properties. It is then called a maguet. 
Such a magnetized steel bar may be used as fundamental in 
the investigation of the properties of magnetism. 

If pieces of iron or steel be brought near a steel magnet, 
they are attracted by it, and unless removed by an outside 
force they remain permanently in contact with it. While in 
contact with the magnet, the pieces of iron or steel also ex- 
hibit magnetic properties. The iron almost wholly loses these 
properties when removed from the magnet. The steel retains 
them and itself becomes a magnet. The reason for this differ- 
ence is not known. It is usually said to be due to a coercive 
force in the steel. The attractive power of the original magnet 
for other iron or steel remains unimpaired by the formation of 
new magnets. 

A body which is thus magnetized or which has its mag- 
netic condition disturbed is said to be affected by magnetic 
tnduction. 


eh 


224 ELEMENTARYS PA YOLCe: [178 


In an ordinary bar magnet there are two small regions, near 
the ends of the bar, at which the attractive powers of the mag- 
net are most strongly manifested. These regions are called 
the poles of the magnet. The line joining two points in these 
regions, the location of which will hereafter be more closely 
defined, is called the magnetic axts. An imaginary plane drawn 
normal to the axis at its middle point is called the equatorial 
plane. 

If the magnet be balanced so as to turn freely in a horizon- 
tal plane, the axis assumes a direction which is approximately 
north and south. The pole toward the north is usually called 
the north or positive pole; that toward the south, the south or 
negative pole. 

If two magnets be brought near together, it is found that 
their like poles repel and unlike poles attract one another. 

If the two poles of a magnet be successively placed at the 
same distance from a pole of another magnet, it is found that 
the forces exerted are equal in amount and oppositely directed. 

The direction assumed by a freely suspended magnet shows. 
that the earth acts as a magnet, and that its north magnetic 
pole is situated in the southern hemisphere. 

If a bar magnet be broken, it is found that two new poles 
are formed, one on each side of the fracture, so that the two. 
portions are each perfect magnets.. This process of making 
new magnets by subdivision of the original one may be, so far 
as known, continued until the magnet is divided into its least 
parts, each of which will be a perfect magnet. 

This last experiment enables us at once to adopt the view 
that the properties of a magnet are due to the resultant action 
of its constituent magnetic molecules. 

178. Law of Magnetic Force.—By the help of the torsion 
balance, the priaciple of which is described in §§ 82, 188, and 
by using verv long, thin, and uniformly magnetized bars, in 
which the polesecan be considered as situated at the extremi- 


178] MAGNETISM. 225 


ties, Coulomb showed that the repulsion between two similar 
poles, and the attraction between two dissimilar poles, is in- 
versely as the square of the distance between them. 

Coulomb also demonstrated the same law by another 
method. He suspended a short magnet so that it could oscil- 
late about its centre in the horizontal plane. He first ob- 
served the time of its oscillation when it was oscillating in. the 
earth’s magnetic field. He then placed a long magnet verti- 
cally, so that one of its poles was in the horizontal plane of 
the suspended magnet, and in the magnetic meridian passing 
through its centre, and observed the times of oscillation when 
the pole of the vertical magnet was at two different distances 
from the suspended magnet. If we represent by 7 the moment 
of inertia of the suspended magnet, by J7/ its magnetic moment, 
by #7 the horizontal intensity of the earth’s magnetism, by 4, 
and #, the force in the region occupied by the suspended mag- 
net due to the vertical magnet in its two positions, it may be shown 
as in § 183 that the times of oscillation of the suspended magnet 


should be respectively ¢7 =z ie bp ict / Ui “TP 
/ is Hare 
NE edit From such equations, by elimination 


(i, + H)M 
the -ratio of 4, and %, was obtained, and was found to be in 
accordance with the law of magnetic force already given. 
All theories of magnetism assume that the force between 
two magnet poles is proportional to the product of the strengths 
of the poles. The law of magnetic force is then the same as 
that upon which the discussion of potential ($$ 28, 29) was 
based. The theorems there discussed are in general applicable 
in the study of magnetism, although modifications in the details 
of their application occur, arising from the fact that the field 
of force about a magnet is due to the combined action of two 
dissimilar and equal poles. | 
If #z and m, represent the strengths of two magnet poles, 7 
15 


226 ' ELEMENTARY PHYSICS. [179 


the distance between them, and # a factor depending on the 
units in which the strength of the pole is measured, the formula 
meme, 


expressing the force between the poles is & 


179. Definitions of Magnetic Ouantitics mm iaw of 
magnetic force enables us to define a uuzt magnet pole, based 
upon the fundamental mechanical units. 

If two perfectly similar magnets, infinitely thin, uniformly 
and longitudinally magnetized, be so placed that their positive 
poles are unit distance apart, and if these poles repel one an- 
other with unit force, the magnet poles are said to be of zzz 
strength. Hence, in the expression for the force between two 


2 


m 
poles, £ becomes unity, and the dimensions of pp are those of 


|= |= MLT-?, 


from which the dimensions of a magnet pole are 


alorce.) sj natas: 


[we] = MtLit - 


This definition of a unit magnet pole is the foundation of the 
magnetic system of units. The strength of amagnet pole is then 
equal to the force which it will exert on a unit pole at unit 
distance. 

The product of the strength of the positive pole of a uni- 
formly and longitudinally magnetized magnet into the distance 
between its poles is called its magnetic moment. 

The quotient of the magnetic moment of such a magnet by 
its volume, or the magnetic moment of unit of volume, is called 
the znztensity of magnetization. 

The dimensions of magnetic moment and of intensity of 


- 380] MAGNETISM. 227 


magnetization follow from these definitions. They are respec 
tively 


[ml |= MiLiT-* and ee = MiL-1T- 


180. Distribution of Magnetism in a Magnet.—lIf we con- 
ceive of a single row of magnetic molecules with their unlike 
poles in contact, we can easily see that all the poles, except 
those at the ends, neutralize one another’s action, and that 
such a row will have a free north pole at one end and a free 
south pole at the other. If amagnet be thought of as made 
up of a combination of such rows of different lengths, the ac- 
tion of their free poles may be seen to be the same as that of 
an imaginary distribution of equal quantities of north and south 
magnetism on the surface and throughout the volume of the 
magnet. If the magnet be uniformly magnetized, the volume 
distribution becomes zero. The surface distribution of magnet- 
ism will sometimes be used to express the magnetization of a 
magnet. In that case what has hitherto been called the mag- 
netic intensity becomes the magnetic density. It is defined as 
the ratio of the quantity of magnetism on an element of sur- 
face to the area of that element. To illustrate this statement, 
we will consider an infinitely thin and uniformly magnetized 


bar, of which the length and cross-section are represented by 


ne ot ae: 
Z ands respectively. Its magnetic intensity is mat ity 


now, for the pole # we substitute a continuous surface distri- 


mL , 
bution over the end of the’ bar, then ~ is also the density of 


that distribution. 
The dimensions of magnetic density follow from this defini- 
tion. ‘They are 


Ee M?* —— Bey 5 gee ae 


225 ELEMEN LAY GaIeY S iC [r8r 


Coulomb showed, by a method of oscillations similar to that 
described in $178, that the magnetic force at different points 
aloug a straight bar magnet gradually increases from the mid- 
dle of the bar, where it is imperceptible, to the extremities. 
This would not be the case if the bar magnet were made up of 
equal straight rows of magnetic molecules in contact, placed 
sidc by side. With such an arrangement there would be no 
force at any point along the bar, but it would all appear at the 
twe ends. The mutual interaction of the molecules of contig- 
uous rows make such an arrangement, however, impossible. 

In the earth’s magnetic field, in which the lines of magnetic 
force may be considered parallel, a couple will be set up on 
any magnet, so magnetized as to have only two poles, due to 
the action of equal quantities of north and south magnetism 
distributed in the magnet. The points at which the forces mak- 
ing up this couple are applied are the fo/es of the magnet, and 
the line joining them is the magnetic axis. These definitions 
are more precise than those which could be given at the outset. 

181. Action of One Magnet on Another.—The investiga- 
tion of the mechanical action of one magnet on another is im- 
portant in the construction of apparatus for the measurement 
of magnetism. 

(1) To determine the potential of a short bar magnet at a 

_p point distant from it, let WS (Fig. 55). 


Me eal if represent the magnet of length 2/, the 

Rese Heal “p ' poles of which are of strength m, and 
ee let the point P be at a distance 7 from. 
Piceice. the centre of the magnet, taken as ori- 


gin. Let the x axis coincide with the axis of the magnet. 
The potential at P is then 


‘3 I ee is 
V = Tea Gea 


I I 
=" EP aay — EP Foal) 


181] MAGNETISM. 229 


This expression expanded gives 


2mlx  3ml’x | 5Sml'x® 


V = —- — —- + | (71) 


1 a 7 


if we assume 7 so large that we may neglect terms of higher 
order inZ. The first term is the most important, and if 7 be 
very great compared with /, the other terms may be neglected. 


The ratio = is the cosine of the angle POW or 6. If we rep- 


resent the magnetic moment 2m/, as is generally done, by J/, 
the potential at any very distant point becomes 


M 
7 cos @. (72) 


Since cos @ is zero for all points in a plane through the ori- 
gin at right angles to the magnetic axis, that plane is an equi- 
potential surface of zero potential. It is the plane defined as 
the equatorial plane. The lines of force evidently originate at 
the poles and pass perpendicularly through this surface. This 
system of lines of force can be easily illustrated by scattering 
fine iron filings on a sheet of paper held over a bar magnet. 
They will arrange themselves approximately along the lines of 
force. 

At a point on the line of the axis where 7 = x, the poten- 
tial becomes 


acest 6 ane (73) 


(2) In one method of application of the instrument called the 
magnetometer it is necessary to know the expression for the 
moment of couple set up bythe action of a magnet at right 


L 


230 ELEMENTARY PAYSICS. (18r 


angles to another, the centre of which is in the prolongation 
of the axis of the first magnet. Let the centre of the first 
magnet be the origin, and its axis the axis. Represent the 
strength of its poles by m, and the strength of the poles of the 
second magnet by m,, the lengths of the two magnets by 2/ 
and 2y respectively. To determine the moment of couple due 
to the action of the first magnet on the second, we must first 
find the component along the 4 axis of the force due to the 
first magnet on a pole m, at a point distant y from the 4 axis. 

m, Lhe force due to the pole of 

|, the first magnet at WV (Fig. 56) 


$ ; ie on a pole 7, is 


mm, 


Fic. 56. y “aie (x iat Z)* 


The cosine of the angle made by this force with the + axis is 
a—Tl 
PRS SA te 


X axis is 


Hence the component of this force along the 


mm (x — L) 
Ge di) 
Hence the component along the x axis of the whole force on 
the pole #,, due to the first magnet, is 


Pia HA atl 
C9) AE 1) ES ta yi) 
When this expression is expanded in increasing negative pow- 


ers of x, neglecting all terms containing higher powers of x 
than the fifth, we obtain 


mm, ( 


182] MAGNETISM. 231 


An-equal and oppositely directed component acts upon the 
other pole — m, of the second magnet. Hence the moment 
of couple due to the action of the first magnet upon the sec- 
ond is 


aero, ‘faery a) 
sin yas A piy E 


(74) 


Pew bersich that 37 — 22, or if the ratio, of the lengths of 


the two magnets used be 1: 71.5, the second and third terms 
vanish, and the expression for the moment of couple depends 
only on the first term of the series. In practice it is not pos- 
sible to completely neglect the other terms, on account of the 
uncertainty as to the position of the poles in the figure of a 
magnet, but by making the lengths of the two magnets as 1 to 


V1.8, the numerator of the term having 2° in the denominator 
is made very small, and is eliminated by the method of obser- 
vation employed, as will be explained in the discussion of the 
magnetometer. 

182. The Magnetic Shell.—A magnetic shell may be de- 
fined as an infinitely thin sheet of magnetizable matter, mag- 
netized transversely ; so that any line in the shell normal to its . 
surfaces may be looked on as an infinitesimally short and thin 
magnet. These imaginary magnets have their like poles con- 
tiguous. The product of the intensity of magnetization at 
any point in the shell into the thickness of the shell at that 
point is called the strength of the shell at that point, and is de- 
noted by the symbol 7. 

The dimensions of the strength of a magnetic shell follow 
at once from this definition. We have [7] equal to the dimen- 
sions of intensity of magnetization multiplied by a length. 
Therefore [7] = 422:T-1, 

We obtain first the potential of such a shell of infinitesi- 


232 ELEMENTARY (PFS SICS. [182 


mal area. Let the origin (Fig. 57) be taken half-way between 
ue the two faces of the shell, and let the 


Ro shell stand perpendicular to the x 

icon iy axis. Let @ represent the area of the 

et fel shell, supposed infinitesimal, 2/7 the 

ol fi------- thickness of the shell, and d the mag. 
Fic. 57. netic intensity. The volume of this’ 


infinitesimal magnet is 2a/, and from the definition of mag- 
netic intensity 2a/d is its magnetic moment. The potential 
at the point P is then given by Eq. 72, since 7 is so small that 
all but the first term in the series of Eq. 71 may be neglected. 
We have 


ee cue i phat cos 6. 
r r 


Now a@ cos @ is the projection of the area of the shell upon a 
plane through the origin normal to the radius vector 7, and, 


We aces 7". 
since @ is infinitesmal, -— is the solid angle @ bounded by 


the lines drawn from P to the boundary of the areaa. The 
potential then becomes V = 2/da = 7, since 2/d is what has 
been called the strength of the shell. ) 

The same proof may be extended to any number of con- 
tiguous areas making upa finite magnetic shell. The potential 
due to such a shell is then 27a. If the shell be of uniform 
strength, the potential due to it becomes 72'@ and is got by 
summing the elementary solid angles. This sum is the solid 
angle Q, bounded by the lines drawn from the point of which 
the potential is required to the boundary of the shell. The 
potential due to a magnetic shell of uniform strength is there- 
fore 


JQ. (75) 


It does not depend on the form of the shell, but only on the angle 


183] MAGNETISM. 233 


subtended by its contour. At a point very near the positive 
face of a flat shell, so near that the solid angle subtended by 
the shell equals 27, the potential is 277, at a point in the plane 
of the shell outside its boundary where the angle subtended is 
zero, the potential is zero; and near the other or negative 
face:of the shell it is — 227. The whole work done, then, in 
moving a unit magnet pole from a point very near one face to 
a point very near the other face is 477. This result is of im- 
poitance in connection with electrical currents. 

183. Magnetic Measurements.—It was shown by Gilbert 
in a work published in 1600, that the earth can be considered 
as a magnet, having its positive pole toward the south and its 
negative toward the north. The determination of the mag- 
netic relations of the earth are of importance in navigation 
and geodesy. The principal magnetic elements are the de- 
clination, the dip, and the horizontal intensity. 

The declination is the angle between the magnetic meridian, 
or the direction assumed by the axis of a magnetic needle 
suspended to move freely in a horizontal plane, and the geo- 
graphical meridian. 

The az is the angle made with the po dened by the axis 
of a magnetic needle suspended so as to turn freely in a verti- 
cal plane containing the magnetic meridian. 

The horizontal intensity is the strength of the earth’s mag- 
netic field resolved along the horizontal line in the plane of the 
magnetic meridian. A magnet pole of strength # in a field in 
which the horizontal intensity is represented by /# is urged 
along this horizontal line with a force equal to mH. From 
this equation the dimensions of the horizontal intensity, and 
so also of the strength of a magnetic field in any case, are 


METS 


ea fae As ML 7. 


— —_ 


234 ELEMENTARY PHYSICS. [183 


The horizontal intensity can be measured relatively to some 
assumed magnet as standard, by allowing the magnet to oscil- 
late freely in the horizontal plane about its centre, and noting 
the time of oscillation. The relation between the magnetic 
moment JZ of the magnet and the horizontal intensity A is 
calculated by a formula. analogous to that employed in the 
computation of g from observations with the pendulum. If 
the magnet be tie displaced from its position of equilib- 
rium, so as to make small oscillations about its point of sus- 
pension, it can be shown as in § 39 that it is describing a simple 
harmonic motion, and as in § 41 (1) that the kinetic energy of 
the magnet when its axis coincides, during an oscillation, with 
the magnetic meridian is 


ang gp 
aL 
4/ — Tia e 


The potential energy at the extremity of its arc is due to the 
magnetic force mH acting on the poles. The component of 
this force which is efficient in moving the magnet is #7 sin @ 
or mHa, if w be always very small. Since a@ varies between o 
and @¢, the average force efficient in turning the needle is 
4mf1p. ‘The poles upon which this force acts move from the 
position of maximum kinetic energy to the position of no 
kinetic energy, through a distance /¢, if / represent the half 
length of the magnet. The potential energy of the couple 
formed by the two poles of the magnet is then mA/¢*, and 
this is equal to the kinetic energy at the point of equilibrium ; 
that is, 


miTlp’. 


Hence if we write 2/— M, the magnetic moment of the mag- 


183] MAGNETISM. |, 235 
‘ Am] st ae at 
net, we obtain ZH = -—;;; or if we take the time of oscillation 
ie 
as ¢ — —, we have 
2 
n° 
COS flim aie (76) 


tw 

The moment of inertia J may be either computed directly 
from the magnet itself, if it be of symmetrical form, or it may 
be determined experimentally by the method of § 36, Eq. 23, 
which applies in this case. The horizontal intensity is then 
determined relative to the magnetic moment of the assumed 
standard magnet. 

This measure may be used to give an absolute measure of 
ff by combining with it another observation which gives an 
independent relation between J7 and A. In one arrangement 
of the apparatus two magnets are used: one, the deflected mag- 
net, so suspended as to turn freely in the horizontal plane; and 
the other, the deflecting magnet, the one of moment J7 used in 
the last operation, carried upon a bar which can be turned 
about a vertical axis passing through the point of suspension 
of the deflected magnet. The centre of the deflected magnet 
is in the prolongation of the axis of the deflecting magnet, and, 
when the apparatus is used, the carrier bar is turned until the 
two magnets are at right angles to one another. The equilib- 
rium established is due to two couples acting on the deflected 
magnet, one arising from the action of the earth’s magnetism, 
and the other from that of the deflecting magnet. This latter 
has been already discussed in § 181. The couple acting on the 


Ve 
deflected magnet is expressed by 4m, y(—s -+- Sah WELE Tene 


represents the small numerator,of the correction term. This 
correction can be made very small in practice by giving to the 


236 ELEMENTARY PHYSICS. [183 


magnets, as already explained, lengths in the ratio of I to V1.5. 
The opposing equal couple is 2#,//y sin ¢, where @ represents 
the angle of deflection from the magnetic meridian. We have 


I lee I PP ¥Aae 
then 4Mmy(* + 5) 27 PLT Stee, OL > + 3 Se P. 
Since P is always a very small quantity, this equation may be 


written 
M P , 
Fag = 42" Sin ot — =). (77) 


P is determined by measuring the angles ¢ and @¢, for two dif. 
ferent distances x and x. The equations containing the results 
of these measurements are 


= a ur (1 — = 


| x 
and 


eee el ean es 
S238 6 sin p(t ea 
aoe J x 


/ 


From these equations the value of P is found to be equal to 


32° sin & — 34,‘ sin @, 
ara Serer ee Ste ee : 
$x¢ sin PY — 94, sin G, 


By substitution of this value of Pin either of the above equa- 


M 
tions, the value of 7a is obtained in absolute units. By com- 
bination of Eq.(76) and Eq. (77) the value of AH is obtained 
independent of JZ, and in absolute units. 

It is evident that the value of M7 can be obtained alse in 


absolute units from the same equations. 


184] MAGNETISM. 237 


In determinations of the horizontal intensity in which great 
accuracy is desired, corrections must be introduced in these 
equations for the changes of magnetic moment due to changes. 
of temperature (§ 185) and to induction (§ 184). 

184. Magnetic Induction.—In the foregoing discussions 
the effect of magnetic induction has been neglected, and the 
magnets considered are those known as permanent magnets. 
Phenomena, however, arise when bodies not permanently mag- 
netized are brought into a magnetic field, which are due to 
magnetic induction. It was found by Faraday that all bodies. 
are affected by the presence of a magnet. Some of them, such 
as iron, nickel, cobalt, and oxygen, seem to be attracted by 
the magnet. Others, such as bismuth, copper, most organic 
substances, and nitrogen, seem to be repelled from the magnet. 
The former are said to be ferromagnetic or paramagnetic; the 
latter, dzamagnetie. 

The most obvious explanation of these phenomena, and the 
one adopted by Faraday, is to ascribe them to a distribution 
of the induced magnetization in paramagnetic bodies, in an 
opposite direction from that in diamagnetic bodies. If a para- 
magnetic body be brought between two opposite magnet poles, 
a north pole is induced in it near the external south pole, and 
a south pole near the external north pole. The magnetic 
separation is then said to be in the direction of the lines of 
force. According to this explanation, then, the separation of 
the induced magnetization in a diamagnetic body is in a direc- 
tion opposite to that of the lines of force. In other words, if 
a diamagnetic body be brought between two opposite magnet 
poles, the explanation asserts that a north pole is induced in it 
near the external north pole, and a south pole near the exter- 
nal south pole. 

One of Faraday’s experiments, however, indicates that the 
different behavior of bodies of these two classes may be due 
only to a more or less intense manifestation of the same action. 


238 ELEMENTARY PHYSICS. [184 


He found that a solution of ferrous sulphate, sealed in a glass 
tube, behaves, immersed in a weaker solution of the same salt, as 
a paramagnetic body; but, when immersed in a stronger solu- 
tion, as a diamagnetic body. It may, from this experiment, be 
concluded that the direction of the induced magnetization is 
the same for all bodies, and that the exhibition of diamagnetic 
or paramagnetic properties depends, not upon the direction of 
induced magnetization, but upon the greater or less intensity 
of magnetization of the surrounding medium. 

Faraday discovered that many bodies, while in a vacuum, 
exhibit diamagnetic properties. In accordance with this ex- 
planation, we must conclude that a vacuum can have magnetic 
properties. It seemed to Faraday unlikely that this should be 
the case, and he therefore adopted the explanation which was 
first given. As it has been since shown that the ether which 
serves as a medium for the transmission of light, and which 
pervades every so-called vacuum, is also probably concerned in 
electrical and magnetic phenomena, there is no longer any 
reason for the opinion that the possession of magnetic proper- 
ties by a vacuum is inherently improbable. In accordance 
with this view, in what follows we shall adopt the second ex- 
planation, which was developed by Thomson. 

In order to express the difference between paramagnetic 
and diamagnetic bodies it is necessary to use some definitions 
which did not appear in the treatment of permanent magnets. 
To understand these it is necessary to determine the magnetic 
force within a magnet, as upon it depends the induced mag- 
netization. The force in the interior of a magnet is measured 
by considering an infinitely short cylinder or thin disk, the 
axis of which is parallel to the axis of the magnet, cut out of 
the interior of the magnet. The force exerted upon a magnet 
pole within this space is the force to be considered. Assume 
a straight bar magnet, uniformly and longitudinally mag- 
netized, and suppose such a disk cut out within it, with faces 


ya 


- 


meay MAGNETISM. 23q 


perpendicular to the magnetic axis. We may then assume 
{§ 180) that there will be a uniform distribution of magnetism 
on both faces of the disk, positive on one face and negative on 
the other. The force due to this imaginary distribution on a 
pole in the centre of the disk will be twice that due to one 
face. If its density be called z, by § 29 (3) the force due 
to one face is 277. If there be besides a magnetic force F in 
the field, the total force on the pole is *+ 472. If the straight 
bar be not originally magnetized, and be placed in a uniform 
magnetic field, it is assumed that z is proportional to /% Let 
z= kf, and call & the coeffictent of tnduced magnetization. We 
have then the total force within the cavity equal to (1+474)F. 
This quantity is called by Maxwell the magnetic tnduction, and 
the factor 1-++- 47k the maguetic inductive capacity oi the sub- 
stance. Thomson and Rowland call it the magnetic permea- 
bility. 

Now, to make clear what is meant by the classification of 
bodies as paramagnetic and diamagnetic, we may proceed as 
follows. Suppose an infinitesimal cube of the substance to be 
tested placed in a magnetic field which is not uniform, with 
one of its faces normal to the lines of force. The intensity of 
the induced magnetization will be £/ if we assume that the 
cube is so small that we may neglect the variation of its in. 
duced magnetization, due to the variation within it of the 
magnetic force /. In most bodies % is so small that we may 
also assume that the induced magnetization does not appre- 
ciably alter the’ field, and that the force and potential at a 
point within the cube are the same asif it werenot there. The 
resultant magnetization is equivalent ($ 180) to a distribution 
of magnetism over the two faces normal to the lines of force, 
with a density equal to £F, positive on the face turned toward 
the positive direction of the lines of force, and negative on the 
other face. Let the length of an edge of the cube be denoted 
by a. Then the quantity of magnetism on each of the two 


240 ELEMENTARY PHYSICS. [184 


faces is kFa*. We are to determine the work done by a move- 
ment of the cube from one point in a magnetic field to another. 
To do this we will determine first the work done by the mag- 
netism on one face of the cube during the movement. 

Consider a series of equidistant points, designated by 
I, 2, 3,...#, ona line of force. Represent by 73) sues 
the potentials of those points, of which V, is the greatest and V,, 
the least value of the potential between the points I and z, and 
suppose the points to be so taken that the differences of poten- 
tial between any two consecutive ones is indefinitely small. The 
work done by a quantity of magnetism equal to £Fa’* in mov- 
ing from the point 1 to the point 2 is kFa*(V,—V,). If F, and 
F, represent the values of / at the points 1 and 2, the average 
force in the distance between them may be set equal to 


FY 
shout —,and the average quantity of induced magnetization on 
He face of the cube considered during this movement is 
phat fa 
aie 
The work done is then— 


Hee 
From 1 to 2, 2’k——— ae —(V, — V,) 
pel ath(n ASA Ee 
2. ve, 2 
FV, RV, Aa 
tra aeia th kis = a(t A 22 _ AP) 
From z — 1 to”, 
or pies Phe lain! Bu Lynas V,, ms ba =), 


184] MAGNETISM. 241 


The work done in moving from the point 1 to the point z 
is the sum of these terms. 
To effect the summation we must show that all terms 


eg 


ac will vanish. 


— 


similar to the two terms rate ate 


The force at the point 2 is the space rate of change of 
potential at that point, taken with the opposite sign. If d 
represent the distance between any two consecutive points, we 

V,— V, 
have /, = — cee, 
V, piss 

A. 

for the sum above mentioned 


in the limit, as d@ approaches zero. So 


also Fy = — - in the limit. Using these values, we have 


EUAN AE | AE Sh aN a 


2 4d 


Now in the limit, as the distances between the points 1, 2, 3, 4, 
memmeaemezerO, we have = V.V,, and V7 = V,V,; hence 
the terms considered and all similar ones vanish. The total 
work done in moving from I to z is 


wil Favs) 
2 2 
_ If we bear in mind our assumption that we may neglect the 
variation of induced magnetism within the cube in any one 
position, we may express the work done by the movement of 
the quantity — ka’F on the opposite face of the cube from a 
point at the distance a measured along a line of torce from the 
point I, to a point similarly situated with respect to the point 
n, by 

— ef LAY) _ Fiat V0) 

2 2 


16 


242 - ELEMENTARY PHYSICS. [184 


In this expression 4V, represents the difference in potential 
between the point I and the point at distance @ from it at 
which the potential is higher than Y,, and 4V, represents a 
similar difference between the point z and the point at distance 
a from it at which the potential is higher than V,,. The work 
done by the whole cube in moving from the point 1 to the 
point z is the sum of the quantities of work done by the quan. 


tities of magnetism on its two faces, and is hence equal to 
ak 
— 7 4 V, — F,4V,). 


From the relation between force and potential we have, in the 


AV. Meg 


limit, as a becomes indefinitely small, 7, = aa and fee : 
a 


since the distance @ is measured in the negative direction. 
Substituting these values, the expression for the work done by 
3 

the cube becomes W = — wee ao) The free movement 
of any system is such as to do work. Hence the cube will 
move from the point 1 along the line of force toward wz if free 
to do so, in case VV is positive. 

Two cases may arise depending on the substance of which 
the cube is composed. We assume the value of & for vacuum 
as zero. If & for any body be positive, the body is paramag- 
netic, and W is >owhen Ff, < #,; the cube moves from a 
place of weaker to a place of stronger magnetic force. If 2 
be negative, the body is diamagnetic, and W is >o when 
Ff, > F,,; the cube moves from a place of stronger to a place 
of weaker magnetic force. 

The subject may be looked at from a different point of 
view. The coefficient of induced magnetization & is negative 
in all diamagnetic bodies, but its numerical value is small. It 


. I . 
has never been found to be numerically greater than ve in 


185] MAGNETISM. 243 


diamagnetic bodies. In such bodies, therefore, the value of y, ° 
the magnetic permeability, is less than 1, though never negative. 
When £ is 0, & equals 1, and for paramagnetic bodies yu is 
greater than 1. The ratio of the force within the substance of 
which the magnetic permeability is “ to that in vacuum, in 


ede 
which it is supposed to be placed, is Grit my ATE Le ret fhe) LE 


the convention of §21 be used, by which the strength of a 
field of force is represented by the number of lines of force 
passing perpendicularly through unit area, it is evident that 
when a paramagnetic body in which “>1 and V> F is 
brought into the field, the lines of force are converged into the 
body. When a diamagnetic body is in the field the lines of 
force are deflected from it. 

As may be easily seen, a paramagnetic body of permea- 
bility ju, surrounded by a medium also paramagnetic, but of 
permeability yu, > ,, will act relative to the medium as a dia- 
magnetic body. The condition of any body of which the 
permeability is less than that of the medium in which it is im- 
mersed is like that of a weak magnet between the ends of two 
stronger ones, all three being magnetized in the same direc- 
tion. The movements of both paramagnetic and diamagnetic 
bodies may be roughly illustrated by the movements of bodies 
immersed in water, which rise or sink according as their specific 
gravities are less or greater than the specific gravity of water. 

185. Changes in Magnetic Moment.—When a magnet- 
izable body is placed in a powerful magnetic field, it often 
receives, temporarily, a more intense magnetization than it can 
retain when removed. It is said to be saturated, or magnetized 
to saturation, when the intensity of its magnetization is the 
greatest which it can retain when not under the inductive 
action of other magnets. The coercive force of steel is much 
greater than that of any other substance; the intensity of 
magnetization which it can retain is, therefore, relatively very - 


244 ELEMENTARY £IPY SICS. [186 


creat, and it is hence used for permanent magnets. It is found 
that the coercive force depends upon the quality and temper 
of the steel. 

Changes of temperature cause corresponding changes in the: 
magnetic moment of a magnet. If the temperature of a mag- 
net be gradually raised, its magnetic moment diminishes by an 
amount which, for small temperature changes, is nearly pro- 
portional to the change of temperature. The magnet recovers. 
its original magnetic moment when cooled again to the initiak 
temperature, provided that the temperature to which it was. 
raised was never very high. If it be raised, however, to a red. 
heat, all traces of its original magnetism permanently disap- 
pear. Trowbridge has shown that, if the temperature of a 
magnet be carried below the temperature at which it was. 
originally magnetized, its magnetic moment also temporarily 
diminishes. 

Any mechanical disturbance, such as jarring or friction,. 
which increases the freedom of motion among the molecules of 
a magnet, in general brings about a diminution of its magnetic 
moment. On the other hand, similar mechanical disturbances. 
facilitate the acquisition of magnetism by any magnetizable 
body placed in a magnetic field. | 

186. Theories of Magnetism.—It has been shown by 
mathematical analysis that the facts of magnetic interactions. 
and distribution are consistent with the hypothesis, which we 
have already made, that the ultimate molecules of iron are 
themselves magnets, having north and south poles which 
attract and repel similar poles in accordance with the law of 
magnetic force. Poisson’s theory, upon which most of the 
earlier mathematical work was based, was that there exist in 
each molecule indefinite quantities of north and south magnetic 
fluids, which are separated and moved to opposite ends of the 
molecule by the action of an external magnetizing force. 
Weber’s view, which is consistent with other facts that Pois— 


186] MAGNETISM. 245 


son’s theory fails to explain, is that each molecule is a magnet, 
with permanént poles of constant strength, that the molecules 
of an iron bar are, in general, arranged so as to neutralize one 
another’s magnetic action, but that, under the influence of an 
external magnetizing action, they are arranged so that their 
magnetic axes lie more or less in some one direction. The bar 
is then magnetized. On this hypothesis there should be a 
limit to the possible intensity of magnetization, which would 
be reached when the axes of all the moiecules have the same 
direction. Direct experiments by Joule and J. Miiller indicate 
the existence of sucha limit. An experiment of Beetz, in which 
a thin filament of iron deposited. electrolytically in a strong 
magnetic field becomes a magnet of very great intensity, points 
in the same direction. The coercive force is, on this hypothesis, 
the resistance to motion experienced by the molecules. The 
facts that magnetization is facilitated by a jarring of the steel 
brought into the magnetic field, that a bar of iron or steel after 
being removed from the magnetic field retains some of its 
magnetic properties, that the dimensions of an iron bar are 
altered by magnetization, the bar becoming longer and dimin- 
ishing in cross-section, and that a magnetized steel bar loses its 
magnetism if it be highly heated, are all facts which are best 


explained by Weber’s hypothesis. 


GHAR DER TE 
ELECTRICITY IN EQUILIBRIUM. 


187. Fundamental Facts.—(1) Ifa piece of glass anda piece 
of resin be brought in contact, or preferably rubbed together, 
it is found that, after separation, the two bodies are attracted 
towards each other. Ifasecond piece of glass and a second 
piece of resin be treated in like manner, it is found that the 
two pieces of glass repel each other and the two pieces of resin 
repel each other, while either piece of glass attracts either piece 
of resin. These bodies are said to be electrified or charged. 

All bodies may be electrified, and in other ways than by 
contact. It is sufficient for the present to consider the single 
example presented. The experiment shows that bodies may 
be in two distinct and dissimilar states of electrification. The 
glass treated as has been described is said to be vitreously or 
positively electrified, and the resin resinously or negatively elec- 
trified. The experiment shows also that bodies similarly elec- 
trified repel one another, and bodies dissimilarly electrified at- 
tract one another. 

(2) If a metallic body, supported on a glass rod, be touched 
by the rubbed portion of an electrified piece of glass, it will 
become positively electrified. If it be then joined to another 
similar body by means of a metallic wire, the second body is at 
once electrified. If the connection be made by means cf a 
damp linen thread, the second body becomes electrified, but not 
so rapidly as before. If the connection be made by means of 
a dry white silk thread, the second body shows no signs of 
electrification, even after the lapse of a considerable time. 
Bodies are divided according as they can be classed with the 


19'7] ELECTRICITY IN EQUILIBRIUM. ~ 247 


metals, damp linen, or silk, as good conductors, poor conductors, 
and izsulators. The distinction is one of degree. All con- 
ductors offer some opposition to the transfer of electrification, 
and no body is a perfect insulator. ; 

A conductor separated from all other conductors by insu- 
lators is said to be zzsulated. A conductor in conducting con- 
tact with the earth is said to be grounded or joined to ground. 

During the transfer of electrification in the experiment 
above described the connecting conductor acquires certain 
properties which will be considered under the head of Electri- 
cal Currents. 

(3) If a positively electrified body be brought near an insu- 
lated conductor, the latter shows signs of electrification. The 
end nearer the first body is negatively, the farther end posi- 
tively, electrified. If the first body be removed, all signs of 
electrification on the conductor disappear. If, before the first 
body is removed, the conductor be joined to ground, the posi- 
tive electrification disappears. If now the connection with 
ground be broken, and the first body removed, the conductor 
is negatively electrified. 

The experiment can be carried out so as to give quantita- 
tive results, in a way first given by Faraday. An electrified 
body, for example a brass ball suspended by a silk thread, is 
introduced into the interior of an insulated closed metallic 
vessel. The exterior of the vessel is then found to be electri- 
fied in the same way as the ball. This electrification disap- 
pears if the ball be removed. If the ball be touched to the 
interior of the vessel, no change in the amount of the external 
electrification can be detected. If, after the ball is introduced 
into the interior, the vessel be joined to ground by a wire, all 
external electrification disappears. If the ground connection 
be broken, and the ball removed, the vessel has an electrifica- 
tion dissimilar to that of the ball. If the ball, after the ground 
connection is broken, be first touched to the interior of the 


248 ELEMENTARY PHYSICS. [187 


vessel and then removed, neither the ball nor the vessel is any 
longer electrified. 

A body thus electrified without contact with any charged 
body is said to be electrified by zzductzon. The above-men- 
tioned facts show that an insulated conductor, electrified by 
induction, is electrified both positively and negatively at once, 
that the electrification of a dissimilar kind to that of the in- 
ducing body persists, however the insulation of the conductor 
be afterwards modified, and that the total positive electrifica- 
tion induced by a positively charged body is equal to that of 
the inducing body, while the negative electrification can ex- 
actly neutralize the positive electrification of the inducing 
body. 

The use of the terms positive and negative is thus justified, 
since they express the fact that equal electrifications of dis- 
similar kinds are exactly complementary, so that, if they be 
superposed on a body, that body is not electrified. These two 
kinds of electrification may then be spoken of as opposite. 

If the glass and resin considered in the first experiment be 
rubbed together within the vessel, and in general if any appa- 
ratus which produces electrification be in operation within the 
vessel, no signs of any external electrification can be detected. 
It is thus shown that, whenever one kind of electrification is 
produced, an equal electrification of the opposite kind is also 
produced at the same time. 

Franklin showed that, by the use of a closed conducting 
vessel of the kind just described, a charged conductor intro- 
duced into its interior and brought into conducting contact 
with its walls is always completely discharged, and the charge 
is transferred to the exterior of the vessel. This procedure 
furnishes a method of adding together the charges on any 
number of conductors, whether they be charged positively or 
negatively. It is thus theoretically possible to increase the , 
charge of such a conductor indefinitely. 


188] ELECTRICITY IN EQUILIBRIUM. 249 


(4) If any instrument for detecting forces due to electrifica- 
tions be introduced into the interior of a closed conductor 
charged in any manner, it ts found that no signs of force due 
to the charge can be detected. The experiment was accurately 
executed by Cavendish, and afterwards tried ona large scale 
by Faraday. It proves that within a closed electrified con- 
ductor there is no electrical force due to the charge on the 
conductor, or that the potential due to the electrical forces is 
uniform within the conductor. 

188. Law of Electrical Force.—If two charged bodies be 
considered, of dimensions so small that they may be neglected 
in comparison with the distance between the bodies, the stress 
between the two bodies due to electrical force is proportional 
directly to the product of the charges which they contain, and 
inversely to the square of the distance between them. 

If Q and Q represent two similar charges, 7 the distance 
between them, and £ a factor depending on the units in which 
the charges are measured, the formula expressing the repulsion 
between them is 


a 


7 


Coulomb used the torsion balance (§ 82) to demonstrate this 
law. At one end ofa glass rod suspended from the torsion 
wire and turning in the horizontal plane is placed a gilded pith 
ball, and through the lid of the case containing the apparatus 
can be introduced a similar insulated ball so arranged that its 
centre is at the same distance from the axis of rotation of the 
suspended system, and in the same horizontal plane, as the 
centre of the first ball. This second ball may be called the 
carrier. 

To prove the law as respects quantities, the suspended ball 
is brought into equilibrium at the point afterwards to be occu- 
pied by the carrier ball. The carrier ball is then charged and 


~ 


250 ELEMENTARY PHYSICS. [189 


introduced into the case. When it comes in contact with the 
suspended ball, tt shares its charge with it and a repulsion 
ensues. The torsion head must then be rotated until the sus- 
pended ball is brought to some fixed point, at a distance from 
the carrier which is less than that which would separate the 
two balls in the second part of the experiment if no torsion 
were brought upon the wire. The repulsion is then measured 
in terms of the torsion of the wire. The charge on the carrier 
is then halved, by touching it with a third similar insulated 
ball, and, the charge on the suspended ball remaining the same, 
the repulsion between the two balls at the same distance is 
again observed. If the case be so large that no disturbing 
effect of the walls enters, and if the balls be small and so far 
apart that their inductive action on one another may be neg- 
lected, the repulsion in the second case is found to be one half 
that in the first case. In general the problem is a far more 
difficult one, for the distribution on the two spheres is not 
uniform. That portion of the distribution dependent on the 
induction of the balls can be calculated, but the irregularities 
of distribution due to the action of the walls of the case and 
other disturbing elements can only be allowed for approxi- 
mately. 

The law as respects distance is proved in a somewhat simi- 
lar way. The repulsions at two different distances are meas- 
ured in terms of the torsion of the wire, the charges on the 
two balls remaining the same. The same corrections must be 
introduced as in the former case. 

189. Distribution.—The law of electrical force has been 
stated in terms of the charges of two bodies. We may, how- 
ever, consider electricity as a quantity which has an existence 
independent of matter and which is distributed in space. The 
fact cited in § 187 (4) shows that this distribution must be 
looked on as being on the surfaces of conductors and not in 
their interiors. If we define surface density of electrification 


190] ELEGI ACL IY: IN LOCILIBRTIUN, 251 


at any point on the surface of a charged conductor as the 
limit of the ratio of the quantity of electricity on an element 
in the surface at that point to the area of the element as that 
area approaches zero, we may measure quantities of electricity 
in terms of surface density. The surface density of electricity 
is usually designated by o. 

If the law of electrical force hold true not only for charges 
on bodies but also for quantities of electricity on the surface 
elements of a conductor, it is evident, from the fact that within 
an electrified conductor there is no electrical force, that its 
surface density of electrification must be proportional at every 
point on its surface to the thickness at that point of a shell of 
matter which is so distributed on that surface that there is no 
force at any point enclosed by the surface. The distribution 
on a charged sphere may, from symmetry, be assumed uniform. 
The fact that there is no electrical force within a charged sphere 
is then, from § 29 (1), consistent with the law of electrical force 
which has been given; and since the means of detecting elec- 
trical force, if there were any, within a charged conductor are 
very delicate, this fact affords a strong corroborative proof of 
the law. 

The determination of the distribution of electricity on irreg- 
ularly shaped conductors is in general beyond our power. If 
we consider, however, a conductor in the form of an elongated 
egg, it can be readily seen that, in order that there may be no 
electrical force within it, the surface density at the pointed end 
must be greater than that anywhere else on its surface. In 
general, the surface density at points on a conducting surface 
depends upon the curvature of the surface, being greater where 
the curvature is greater. Thus, if the conductor be a long rod 
terminating in a point, the surface density at the pointed end 
is much greater than that anywhere else on the rod. 

190. Unit Charge.—The law of electrical force enables us 
to define a uzzt charge, based upon the fundamental mechanical 
units. 


252 LYON EM LAR Y. CRIS Gas [191 


Let there be two equal and similar positive charges concen- 
trated at points unit distance apart in air, such that the repul- 
sion between them equals the unit of force. ‘Then each of the 
charges is a unit charge, or a unit quantity of electricity. With’ 
this definition of unit charge, it may be said that the force be- 
tween two charges is not merely proportional to, but equals, 
the product of the charges divided by the square of the dis- 
tance between them. The factor £ in the expression for the 
force between two charges becomes unity, and the dimensions 


Q, 


of — are those of a force. If the charges be equal, we have 


BE _— YET. 


r 


Hence [Q] = /?L'T ~* are the dimensions of the charge. This 
equation gives the charge in absolute mechanical units, and 
by means of it all other electrical quantities may be expressed 
in absolute units. It is at the basis of the electrostatic system 
of electrical measurements. 

The practical unit of charge or quantity is called the cou- 
fomb. Itis the quantity of electricity transferred during one 
second by a current of one ampere (§ 218). 

191. Electrical Potential.— The electrical forces have a po- 
tential similar to that discussed in § 28. The unit quantity ef 
positive electricity is taken as the test unit. Since [§ 187 (4)] 
the potential at every point of a charged conductor is the 
same, the surface of the conductor is an equipotential surface. 
The potential of this surface is often called the potentzal of the 
conductor. A conductor joined to ground is at the potential 
of the earth. It will be shown (§ 195) that the potential of 
the earth is not appreciably modified when a charged conduc- 
tor is joined to ground. 


I9gt| ELECTRICITY AN EQUILIBRIUM. 253 


For these reasons it is usual to take the potential of the 
earth as the fixed potential or zero from which to reckon the 
potentials of electrified bodies. The potential of a freely 
electrified conductor and of the region about it is thus positive 
when the charge of the conductor is positive, and negative 
when it is negative. A conductor joined to ground is at 
zero potential. : 

ihe difference of potential between two points is equal to 
the work done in carrying a unit quantity of electricity from 
one point to another. We then have the equation Q(V, — V) 
= work. Hence follows the dimensional equation [V, — V] = 

Ap at as 
LS hai 
tial in electrostatic units. 

If any distribution of a charge exist on a conductor, which is. 
such that the potential at all points in the conductor is not the 
same, it is unstable, and a rearrangement goes on until the po- 
tential becomes everywhere the same. The process of rear- 
rangement is said to consist in a flow of electricity from points 
of higher to points of lower potential. 

On this property of electricity depends the fact that a 
closed conducting surface completely screens bodies within it 
from the action of external electrical forces. For, whatever 
changes in potential occur in the region outside the closed con- 
ductor, a redistribution will take place in it such as to make the 
potential of every point within it the same. Electrical force 
depends on the space rate of change of potential, and not on its. 
absolute value. Hence the changes without the closed conductor 
will have no effect on bodies within it. Further, any electrical 
operations whatever within the closed conductor will not change 
the potential of points outside it. For, whatever operations 
go on, equal amounts of positive and negative electricity always 
exist within the conductor, and hence the potential of the con- 
ductor remains unaltered. Hence electrical experiments per- 


= M'*L*T ~", the dimensions of difference of poten- 


254 ELEMENTARY PEEVSICS. [ror 


formed within a closed room yield results which are as valid as 
if the experiments were performed in free space. | 

The advantage gained by the use of the idea of potential 
in discussions of electrical phenomena may be illustrated by a 
statement of the process of charging a conductor by induction 
described in § 187 (3). To fix our ideas, let us suppose that 
the field of force is due to a positively electrified sphere, and 
that the body to be charged is a long cylinder. When this 
cylinder, previously in contact with the earth and therefore at 
zero potential, is brought end on to a point near the sphere, it 
is in a region of positive potential, and is itself at a positive 
potential. If we consider the original potentials at the points 
in the region now occupied by the cylinder, it is easily seen 
that the potential of points nearer the sphere was higher than 
that of those more remote. When the cylinder is brought into 
the field, therefore, the portion nearer the sphere is temporarily 
raised to a higher potential than the portion more remote. 
The difference of potential between these portions is annulled 
by a flow of electricity from the points of higher potential to 
those of lower potential at a rate depending on the conductiv- 
ity of the cylinder. The end of the cylinder nearer the sphere 
is negatively charged, the end more remote is positively 
charged, and the two charged portions are separated by a line 
on the surface, called the neutral line, on which there is no 
charge. 

If the cylinder be now joined to ground, a flow of electricity 
takes place through the ground connection, and it is brought 
to zero potential. The potential of the cylinder is therefore 
everywhere lower than the original potentials of the points in 
the region which it occupies. This necessitates a negative charge 
distributed over the whole cylinder. In other words, the earth 
and the cylinder may be considered as forming one conductor 
charged by induction, in which the neutral line is not within 
the cylinder. 


192] ELECTRICITY IN EQUILIBRIUM. | 255 


If the ground connection be broken the electrical relations 
are not disturbed. If the cylinder be now removed to a region 
of lower potential against the attraction of the sphere, work 
will be done against electrical forces, which reappears as electri- 
cal energy. The potential of the cylinder is lowered, and, if it 
be again connected with the earth, work will be done by a flow 
of electricity to it. ; 

The fact that there is no electrical force within a closed 
electrified conductor of any shape permits some extensions of 
the theorems of § 209. 

Some small portion of the surface of any electrified conduc- 
tor may be considered a plane relatively to a point situated 
just outside it. Represent the surface density of electricity on 
that plane by co. It was proved (§ 29) that the force due to 
such a plane is 270, if we substitute o for the corresponding 
factor d. Now, just inside the conductor the force is zero. | 
This results from the equilibrium of the force due to the plane 
portion and that due to the rest of the conductor. The force 
due to the rest of the conductor is therefore 270. Ata point 
just outside the conductor these two forces act in the same 
direction. Hence the total force due to the conductor ata 
point just outside it is the sum of the two forces, or 470. 

From the preceding proposition follows at once a deduction 
as to the pressure outwards on the surface of an electrified 
conductor due to the repulsion of the various parts of the 
charge for one another. Select any small portion of the sur- 
face of the electrified conductor of areaa. The force on unit 
quantity acting outward from the conductor at a point in that 
area due to the charge of the rest of the conductor is2zo0. This 
force acts on every unit of charge on the area. The force on 
the area acting outwards is then 27ao”, or the pressure at a 
point in the area referred to unit of area is 270°. This quan- 
tity is often called the electric pressure. 

192. Capacity.—The electrical capacity of a conductor is 
defined to be the charge which the conductor must receive to 


256 ELEMENTARY PHYSICS. [192 


raise it from zero to unit potential, while all other conductors 
in the field are kept at zero potential. This charge varies for 
any one conductor in a way which cannot be always definitely — 
determined, depending upon the medium in which the con- 
ductor is immersed and the position of other conductors in 
the field. When the charged conductor is in very close prox- 
imity to another conductor which is kept at zero potential, the 
amount of charge needed to raise it to unit potential is very 
great as compared with that required when the other conduc- 
tor is more remote. Such an arrangement is called a condenser. 
If the charge on a conductor be increased, the increase in po- 
tential is directly as that of the charge. Hence the capacity 
C is given by dividing any given charge on a conductor by the 
potential of that conductor, or 


O 


The practical unit of capacity is the farvad, which is the ca- 
pacity of a conductor, the charge on which is one coulomb | 
($ 190) when its potential is one volt (§ 228). This unit is too 
great for convenient use. Instead of it a mzcrofarad, or the 
one-millionth part of a farad, is usually employed. 

The equation gives the dimensions of capacity. Measured 
in electrostatic units, they are 


QO} MtLiT-: 
l= || = gam = 2 


Capacity, therefore, is of the dimensions of a length. 

In the theory of Faraday, which has been adopted and de- 
veloped by Maxwell, electrification is made to consist in an 
arrangement or displacement of the insulating medium, called 
by him the d@electric, surrounding the electrified conductor. 


193] BLEGIRICLLY IN LOGTLIBRIGM, 257 


This displacement, beginning at the surface of the electrified 
conductor, continues throughout the dielectric until it termi- 
nates at the surfaces of other conductors. ‘The electrification 
of the charged conductor is the manifestation of this displace- 
ment at one face of the dielectric, that of the surrounding con- 
ductors the manifestation of the displacement on the other 
face. The one charge cannot exist without an equal and op- 
posite charge on surrounding conductors, as was experiment- 
ally proved by Faraday’s experiment already described in 
§ 187 (3). It is therefore necessary, in considering the capacity 
of any conductor, to take account of the medium in which it 
is immersed, and of the arrangement of surrounding conductors. 

193. Specific Inductive Capacity.— The fact that the 
capacity of a condenser of given dimensions depends upon the 
medium used as the dielectric was first discovered by Caven- 
dish, and afterwards rediscovered by Faraday. The property 
of the medium upon which this fact depends is called its 
specific inductive capacity. The specific inductive capacity of 
vacuum is taken as the standard. If Q represent the charge 
required to raise a condenser in which the dielectric is vacuum 
toa potential VY, then if another dielectric be substituted for 
vacuum, it is found that a different charge Q, is required to 


vines oe 
raise the potential to V. The ratio vo) = £K is the specific in- 


‘ Ob 


. : Q aa 
ductive capacity. Since C,= a aPC Cpe 7a are the capacities 


of the condenser with the two dielectrics, it follows that 
C, = CK, (79) 


where C is the capacity with vacuum as the dielectric. The 

specific inductive capacity K is always greater than unity. 

Seme dielectrics, such as glass and hard rubber, have a high 
17 


258 ELEMENTARY PHYSICS. [194 


specific inductive capacity, and at the same time are capable 
of resisting the strain put upon them by the electric displace- 
ment to a much greater extent than such dielectrics as air. 
They are therefore used as dielectrics in the construction of 
condensers. 

194. Condensers.—The simplest condenser, one which ad- 
mits of the direct calculation of its capacity, and from which 
the capacities of many other condensers 
may be approximately calculated or in- 
ferred, consists of a conducting sphere 
surrounded by another hollow concentric 
conducting sphere which is kept always 
at zero potential by a ground connection. 
For convenience we assume the specific 

Tiers inductive capacity of the dielectric sepa- 
rating the spheres to be unity. Let the radius of the small 
sphere (Fig. 58) be denoted by R, that of the inner spherical 
surface of the larger one by 7; let a charge Q be given to the 
_ inner sphere by means of a conducting wire passing through 
an opening in the outer sphere, which may be so small as to 
be negligible. This charge Q will induce on the outer sphere 
an equal and opposite charge, — Q. Since the distribution on 
the surface of the spheres may be assumed uniform, the poten- 
tial at the centre of the two spheres, due to the charge on the 


inner one, is and the potential due to the charge of the 


outer sphere is — ey Hence the actual potential V at the 
/ 


centre, due to both charges, is 


194] . BLECTRICITY IN ZOUCILIBRIUNM, 259 


Hence the capacity is 
Q 
GAR i) 


In order to find the effect of a variation of the value of R, 
divide numerator and denominator by #, and write 


Now, if &, be greater than FR by an infinitesimal, the fraction 
“ is less than unity by an infinitesimal, and the capacity of 
the accumulator is infinitely great. It becomes infinitely small 
if L be diminished without limit. The presence of any finite 
charge at a point would require an infinite potential at that 
point, which is of course impossible. The existence of finite 
charges concentrated at points, which we have assumed some- 
times in order to more conveniently state certain laws, is 
therefore purely imaginary. If electricity is distributed in 
space, it is distributed like a fluid, a finite quantity of which 
never exists at a point. 

If A, increase without limit, C becomes more and more 
nearly equal to R. Suppose the inner sphere to be surrounded 
not by the outer sphere but by conductors disposed at unequal 
distances, the nearest of which is still at a distance FX, so great 


R ; 
that = may be neglected in comparison with unity. Then if 


ia 
the nearest conductor were a portion of a sphere of radius X, 
concentric with the inner sphere, the capacity of the inner 
sphere would be approximately Rk. And this capacity is evi- 


dently not less than that which would be due to any arrange- 


260 ELEMENTARY PHYSICS. [194 


ment of conductors at distances more remote than &. There- 
fore the capacity of a sphere removed from other conductors 
by distances very great in comparison with the radius of the 
sphere is equal to its radius &. This value & is often called 
the capacity of a freely electrified sphere. Strictly speaking, a 
freely electrified conductor cannot exist; the term is, however, 
a convenient one to represent a conductor remote from all 
other conductors. 

A common form of condenser consists of two flat conduct- 
ing disks of equal area, placed parallel and opposite one 
another. The capacity of such a condenser may be calculated 
from the capacity of the spherical condenser already discussed. 
Let d represent the distance 2, — & between the two spherical 
surfaces. Let A and A, represent the area of the surfaces of 
the two spheres of radius R and &,. Then we have 


aes and pe ets 
An 


4 
The capacity of the spherical condenser may then be written 
V AA, 
4nd ~ 
If R, and & increase indefinitely, in such a manner that R, —R 
always equals d, in the limit the surfaces become plane and 4 


becomes equal to A,. The capacity therefore equals Pal 


Since the charge is uniformly distributed, the capacity of any 
portion of the surface cut out of the sphere is proportional to 
the area S of that surface, or 


Ce (81) 


195] ELECTRICITY IN EQUILIBRIUM. 261 


This value is obtained on the assumption that the distribution 
over the whole disk is uniform, and the irregular distribution 
at the edges of the disk is neglected. It is therefore only an 
approximation to the true capacity of such a condenser. 

The so-called Leyden jar is the most usual form of con- 
denser in practical use. It is a glass jar coated with tinfoil 
within and without, up to a short distance from the opening. 
Through the stopper of the jar is passed a metallic rod fur- 
nished with a knob on the outside and in conducting contact 
with the inner coating of the jar. To charge the jar, the outer 
coating is put in conducting contact with the ground, and the 
knob brought in contact with some source of electrification. 
It is discharged when the two coatings are brought in conduct- 
ing contact. When the wall of the jar is very thin in compari- - 
son with the diameter and with the height of the tinfoil coat- 
ing, the capacity of the jar may be inferred from the preceding 
propositions. It is approximately proportional directly to the 
coated surface, to the specific inductive capacity of the glass, 
and inversely to the thickness of the wall. 

195. Systems of Conductors.—lIf the capacities and poten- 
tials of two or more conductors be known, the potential of the 
system formed by joining them together by conductors is easily 
found. It is assumed that the connecting conductors are fine 
wires, the capacities of which may be neglected. Then the 
charges of the respective bodies may be represented by C,V,, 
C,V,,.'.- C,V,, and the capacity of the system by the sum 
c¢,+c,+...C,. Hence V, the potential after connections 
have been made, is 


In the case of two freely electrified spheres joined up 


262 ELEMENTARY PHYSICS. [196 


together by a fine wire, we have C, = &,, and C, = &,, where RX, 
and &, represent the radii of the spheres. Hence we have 


7 RAR, 
ear 


When AR, is very great compared with &,, we obtain 


2 


is R 
Unless V, is so great that the term RM: becomes appreci- 


able, the potential of the system is appreciably equal to the 
original potential of the larger sphere. Manifestly the same 
result follows if A, represent the capacity of any conductor 
relatively small compared with the capacity of the large sphere. 
This proposition justifies the adoption of the potential of the 
earth as the standard or zero potential. 

196. Energy of Charge.—In order to find the work done 
in charging a conducting’ body to a given potential, we will 
consider all surrounding bodies as being kept by ground con- * 
nections at zero potential. Then if an infinitesimal charge be 
given to the body, previously uncharged and at zero potential, 
the work done is that which would be done if the charge were 
brought from infinity to a point of potential o; that is, the work 
=o. The charge g raises the potential of the body so that it 
becomes v, = A: If then another infinitesimal charge g be 


given to the body, the work done is equal to gv, or os and the 


SUS Nie ate 2 
potential is raised to v, = -! So also the work done when 


CG. 


197] ELECTRICITY IN EQUILIBRIUM. —~ 263 


the (z-++7)th charge is given to the body is gv,, and the 


potential becomes wpe The total work done is then 


W=qv,+ 4, 4...) = SO $24.00) 
(z-- 1)n ¢ 107 


2 a 


(82) 


where mg = Qand V=v,. When the charges g are infinitesi- 
mal, Q is equal to the sum of all the charges given to the body. 
Hence the work done in raising a body from zero potential to 
potential V is equal to one half the charge multiplied by the 
potential of the body. 

197. Strain in the Dielectric.—An instructive experiment 
illustrating Faraday’s theory that the electrification of a con- 
ductor is due to an arrangement in the dielectric surrounding 
it, may be performed with a jar so constructed that both coat- 
ings can be removed from it. If the jar be charged, the coat- 
ings removed by insulating handles without discharging the 
jar, and examined, they will be found to be almost without 
charge. If they be replaced, the jar will be found to be charged 
as before. The jar will also be found to be charged if new 
coatings similar to those removed be put in their place. This 
result shows that the true seat of the charge is in the dielectric. 
The experiment is due to Franklin. 

That the arrangement in the dielectric is of the nature of a 
strain is rendered probable by the fact, first noticed by Volta, 
that the volume occupied by a Leyden jar increases slightly 
when the jar is charged. Similar changes of volume were ob- 
served by Quincke in fluid dielectrics as well as in different 
solids. 

Another proof of the strained condition of dielectrics is 
found in their optical relations. It was discovered by Kerr 


204 ELEMENTARY PHYSICS. [197 


that dielectrics previously homogeneous become doubly re- 
fracting when subjected toa powerful electrical stress. Max- 
well has shown, from the assumptions of his electromagnetic 
theory of light, that the index of refraction of a transparent 
dielectric should be proportional to the square root of its 
specific inductive capacity. Numerous experiments, among 
which those of Boltzmann on gases are the most striking, show 
that this predicted relation is very close to the truth. 

It has further been shown that the specific inductive capac- 
ity of sulphur has different values along its three crystallo- 
graphic axes. This is probably true also for other crystals. 

Some crystals, while being warmed, exhibit on their faces 
positive and negative electrifications, which are reversed as 
the crystals are cooling. This fact, while as yet unexplained, 
is probably due to temporary modifications of molecular ar- 
rangement by heat. 

If a jar be discharged and allowed to stand for a while, a 
second discharge can be obtained from it. By similar treat- 
ment several such discharges can be obtained in succession. 
The charge which the jar possesses after the first discharge is 
called the reszdual charge. It does not attain its maximum 
immediately, but gradually, after the first discharge. The 
attainment of the maximum is hastened by tapping on the 
wall of the jar. This phenomenon was ascribed by Faraday to 
an absorption of electricity by the dielectric, but this explana- 
tion is at variance with Faraday’s own theory of electrification. 
Maxwell explains it by assuming that want of homogeneity in 
the dielectric admits of the production of induced electrifica- 
tions at the surfaces of separation between the non homogene- 
ous portions. When the jar is discharged the induced electri- 
fications within the dielectric tend to reunite, but, owing to 
the want of conductivity in the dielectric, the reunion is 
sradual. After a sufficient time has elapsed, the alteration of — 
the electrical state of the dielectric has proceeded so far as to 


198] ELECTRICITY IN EQUILIBRIUM. 265 


sensibly modify the field outside the dielectric. The residual 
charge then appears in the jar. 

198. Electroscopes and Electrometers.—An electroscope 
is an instrument to detect the existence of a difference of electri- 
cal potential. It may also give indications of the amount of 
difference. It consists of an arrangement of some light body 
or bodies, such as a pith ball suspended by a silk thread, ora 
pair of parallel strips of gold-foil, which may be brought near 
or in contact with the body to be tested. The movements of 
the light bodies indicate the existence, nature, and to some 
extent the amount of the potential difference between the body 
tested and surrounding bodies. 

An electrometer is an apparatus which gives precise measure- 
ments of differences of potential. The most important form 
is the absolute or attracted disk electrometer, originally devised 
by Harris, and improved by Thomson. The essential portions 


of the instrument (Fig. 59) are a large 

flat disk B which can be put in con- cc 2 A 
ducting contact with one of the two - 
bodies between which the difference of Fico, 


potential is desired; a similar disk C, in the centre of which 
is cut a circular opening, placed parallel to and a little distance 
above the former one; a smaller disk A with a diameter a little 
less than that of the opening, which can be placed accurately 
in the opening and brought plane with the larger disk; and 
an arrangement, either a balance arm or a spring of known 
strength, from which the small disk is suspended, and by 
means of which the force acting on the disk when it is plane 
with the surface of the larger disk can be measured. The 
three disks can be conveniently styled the attracting disk, the 
guard ring,and the attracted disk. The position of the at- 
tracted disk when it is in the plane of the guard ring is often 
called the sighted position. The guard ring is employed in order 
that the distribution on the attracted disk may be uniform. 


266 ELEMENTARY PHYSICS. [198 


To determine the difference of potential between the at- 
tracted and attracting disks, we consider them first as forming 
a flat condenser. If we represent by Q the quantity of elec- 
tricity on the attracted disk, by / and V, the potentials of the 
attracted and attracting disks respectively, by d the distance 
between them, and by S the area of the attracted disk, then, 
as has been shown in § 194, the capacity of such a condenser is 


La keen 
V,—V~ gna’ 


Now from the nature of the condenser, and in consequence of 
the regular distribution due to the presence of the guard ring, 


OQ 
we have > =o, the surface density on either plate, whence 


—V 

ARAM 
must be eliminated by means of an equation obtained by ob- 
servation of the force with which the two disks are attracted. 
The plates are never far apart, and the force on a unit charge 
due to the charge on the lower one may be always taken in 
the space between the plates as equal to 270 (§ Ig1). Every 
unit on the attracted disk is attracted with this force, and the 
total attraction, which is measured by means of the balance or 
spring, is # = 27z0°S. Substituting this value of o in the for- 
mer equation, we get 


The surface density cannot be measured, and 


_—_— 


Vii ee (83) 


Cd 


which gives the difference of potential between the two plates 
in terms which are all measurable in absolute units. In Thom- 


198] LLEGCLRIGLT, YoiN £OCUILIBRIOM. 267 
son’s form of the electrometer the attracted disk is kept at a 
high constant potential Y; the attracting disk is brought to 
the potential V, of one of the two bodies of which the differ- 
ence of potential is desired, and the position of the attracting 
disk when the attracted disk is in its sighted position is noted. 
The attracting disk is then brought to the potential V, of the 
other body, and by a micrometer screw the distance is measured 
through which the attracting disk is moved in order to bring 
the attracted disk again into its sighted position. This meas- 
urement can be made with much greater precision than the 
measurement of the distance between the two plates. The 
formula is easily deduced from the one already given. Inthe 
first observation we have 


V,—Vaay 2, 


in the second, 


whence 


V,—V,= (4, — a 5, (84) 


and d, — d, is the distance measured. 

Thomson’s guadrant electrometer is an instrument which is 
not used for absolute measurement, but being extremely sensi- 
tive to minute differences of potential, it enables us to compare 
them with each other and with some known standard. The 
construction of the apparatus can best be understood from 


268 ELEMENTARY PHYSICS. [199 


ee ree 


Fig. 60. Of the four metallic quadrants which are mounted 
“on insulating supports, the two marked P 
and the two marked JV are respectively in 
conducting contact by means of wires. 
The body C, technically called the needle, 
isa thin sheet of metal, suspended sym- 
metrically just above the quadrants by 
two parallel silk fibres, forming what is 
Fi. 60. known asa bifilar suspension. When there 
is no charge in the apparatus, the axes of symmetry of the 
needle lie above the spaces which separate the quadrants. 

To use the apparatus, the needle is maintained at a high, 
constant potential, and the two points, the difference of poten- 
tial between which is desired, are joined to the pairs of quad- 
rants Pand V. The needle is deflected from its normal posi- 
tion, and the amount of deflection is an indication of the 
difference of potential between the two pairs of quadrants. 

199. Electrical Machines.—£/ectrical machines may be 
divided into two classes: those which depend for their opera 
tion upon friction, and those which depend upon induction. 

The frictional machine, in one of its forms, consists of a 
circular glass p/ate, mounted so that it can be turned about an 
axis, and a rubber of leather, coated with a metal amalgam, 
pressed against it. The rubber is mounted on an insulating 
support, but, during the operation of the machine, it is usually 
joined to ground. Diametrically opposite is placed a row o1 
metal points, fixed in a metallic support, constituting what is 
technically called the comé. The comb is usually joined to an 
accessory part of the machine presenting an extended metallic 
surface, called the przme conductor. The prime conductor is 
carried on-an insulating support. | 

*When the plate is turned, an electrical separation is pro- 
duced by the friction of the rubber, and the rubbed portion of 
the plate is charged positively. When the charged portion of 


199] ELECTRICITY IN EQUILIBRIUM. 269 


the plate passes before ine comb, an Aeris aes OC- 
curs in the prime conductor due to the inductive action of the 
plate, a negative charge passes from the comb to neutralize 
the positive charge of the plate, and the prime conductor is 
charged positively. Since accessions are received to the charge 
of the prime conductor as each portion of the plate passes the 
comb, it is evident that the potential of the prime conductor 
will continuously rise, until it is the same as that of the plate, 
or until a discharge takes place. 

The fundamental operations of all zxduction machines are 
presented by the action of the e/ectrophorus, an instrument in- 
vented by Voltain 1771. It consists of a plate of sulphur or 
rubber, which rests on a metallic plate, and a metallic disk 
mounted on an insulating handle. The sulphur is electrified 
negatively by friction, and the disk, placed upon it and joined 
to ground, is charged positively by induction. When the 
ground connection is broken and the disk lifted from the 
sulphur, its positive charge becomes available. The process is 
precisely similar to that described in $191. It may evidently 
be repeated indefinitely, and the electrophorus may be used as 
a permanent source of electricity. 

It is evident that a charged metallic piate may be substi- 
tuted for the sulphur in the 
construction of an electropho- 
rus, provided that the disk be 
not brought in contact with it, 
but only near it. A plan by 
which this is realized, and at 
the same time an imperceptible 
charge on one plate is made to 
develop an indefinite quantity 
of electricity of high potential, 
is shown in Fig. 61. A, and 
A, are conducting plates, called zxductors. In front of them 


270 ELEMENTARY PHYSICS. [199 


two disks 4, and &,, called carrzers, are mounted on en arm so 
as to turn about the axis &. Projecting springs 4, and J@, at- 
tached to these disks are so fixed as to touch successively the 
pins D, and D,, connected with the plates A, and A,, and the 
pins C, and C,, insulated from the plates, but joined to the 
prime conductors /, and /. | 7 
"Suppose the prime conductors to be in contact and the car- 
riers so placed that 4, is between DY, and C,, and suppose the 
plate A, to be at a slightly higher potential than the rest of 
the machine. The carrier 4, is then charged by induction. 
When the carriers are turned in the direction of the arrows, and 
the carrier B, makes contact with the pin C,, it loses a part of 
its positive charge and the prime conductors become positively 
charged. At the same time the carrier 5, becomes positively 
charged. As the carrier 4, passes over the upper part of the 
plate A,, the lower part of the plate A, is charged positively 
by induction. This positive charge is neutralized by the nega- 
tive charge of the carrier 4,, when contact is made at D,. The 
plate A, is then negatively charged. The carrier 2, at its con- 
tact at D, shares its positive charge with the plate 4,.. The 
carriers then return to the positions from which they started, 
and the difference of potential between the plates A, and J, is 
greater than it was at first. When, after sufficient repetition 
of this process, the difference of potential has become sufh- 
ciently great, the prime conductors may be separated, and 
the transfer of electricity between the points 7, and /, then 
takes place through the air. Obviously the number of carriers 
may be increased, with a corresponding increase in the rapidity 
of action of the machine. This improvement is usually effect- 
ed by attaching disks of tin-foil at equal distances from each 
other on one face of a glass wheel, so that, as the wheel re- 
volves, they pass the contact points in succession. 

Another induction machine, invented by Holtz, differs in 
plan from the one just described in that the metallic carriers 


199] ELECTRICITY IN EQUILIBRIUM. ant 


are replaced by a revolving glass plate, and the two metallic 
inductor plates, by a fixed glass plate. In the fixed plate are 
cut two openings, diametrically opposite. Near these open- 
ings, and placed symmetrically with respect to them, are fixed 
upon the back of the plate two paper sectors or armatures, 
terminating in points which project into the openings. In 
front of the revolving plate and opposite the ends of the arma- 
tures nearest the openings are the combs of two prime con- 
ductors. Opposite the other ends of the armatures, and also 
in front of the revolving wheel, are two other combs joined to- 
gether by a cross-bar. 

In order to set this machine in operation, one of the paper 
armatures must be charged from some outside source. The 
surface of the revolving plate performs the functions of the 
carriers in the induction machine already explained. The 
armatures take the place of the inductors, and the points in 
which they terminate serve the same purpose as the contact 
points in connection with the inductors. The explanation of 
the action of this machine is, in general, similar to that already 
given. The effect of the combs joined by the cross-bar is 
equivalent to joining to ground that portion of the outside 
face of the revolving plate which is passing under them. 


GHAPBER TT 
THE ELECTRICAL CURRENT. 


200. Fundamental Effects of the Electrical Current.— 
In 1791 Galvani of Bologna published an account of some 
experiments made two years before, which opened a new de- 
partment of electrical science. He showed that, if the lumbar 
nerves of a freshly skinned frog be touched by a strip of metal 
and the muscles of the hind leg by a strip of another metal, 
the leg is violently agitated when the two pieces of metal are 
brought in contact. Similar phenomena had been previously 
observed, when sparks were passing from the conductor of an 
electrical machine in the vicinity of the frog preparation. 

He ascribed the facts observed to a hypothetical animal 
electricity or vital principle, and discussed them from the 
physiological standpoint; and thus, although he and his im- 
mediate associates pursued his theory with great acuteness, 
they did not effect any marked advance along the true direc- 
tion. Volta at Pavia followed up Galvani’s discovery in a 
most masterly way. He showed that, if two different metals, 
or, in general, two heterogeneous substances, be brought in 
contact, there immediately arises a difference of electrical po- 
tential between them. He divided all bodies into two classes. 
Those of the frst class, comprising all simple bodies and many 
others, are so related to one another that, if a closed circuit be 
formed of them or any of them, the sum of all the differences 
of potential taken around the circuit in one direction is equal 
to zero. Ifa body of the second class be substituted for one of 


200] tHE LLECTRICAL CURRENT. 273 


the first class, this statement is no longer true. There exists 
then in the circuit a preponderating difference of potential in 
one direction. Volta described in 1800 his famous voltaic 
battery. He placed in a vessel, containing a solution of salt 
in water, plates of copper and zinc separated from one another. 
When wires joined to the copper and zinc were tested, they 
were found to be at different potentials, and they could be 
used to produce the effects observed by Galvani. The effects 
were heightened, and especially the difference of potential be- 
tween the two terminal wires was increased, when several such 
cups were used, the copper of one being joined to the zinc of 
the next so as to forma series. This arrangement was called 
by Volta the galvanic battery, but is now generally known as 
the voltaic battery. 

Volta observed that, if the terminals of his battery were 
joined, the connecting wire became heated. 

Soon after Volta sent an account of the invention of his 
battery to the Royal Society, Nicholson and Carlisle observed 
that, when the terminals of the battery were joined by a 
column of acidulated water, the water was decomposed into 
its constituents, hydrogen and oxygen. 

In 1820 Oersted made the discovery of the relation e- 
tween electricity and magnetism. He showed that a magnet 
brought near a wire joining the terminals of a battery is de- 
flected, and tends to stand at right angles to the wire. His 
discovery was at once followed up by Ampére, who showed 
that, if the wire joining the terminals be so bent on itself as to 
form an almost closed circuit, and if the rest of the circuit be 
so disposed as to have no appreciable influence, the magnetic 
potential at any point outside the wire will be similar to that 
due to a magnetic shell. 

In 1834 Peltier showed that, if the terminals of the battery 
be joined by wires of two different metals, there is a produc- 


tion or an absorption of heat at the point of contact of the 
18 


274 ELEMENTARY PHYSICS. [201 


wires, depending upon which of the wires is joined to the ter- 
minal the potential of which is positive with respect to the 
other. This fact is referred to as the Peltier effect. 

201. Electromotive Force.—In 1833 Faraday showed con- 
clusively that if a Leyden jar be discharged through a circuit, 
it will produce the same thermal, chemical, and magnetic 
effects as those just described as produced by the voltaic 
battery. 

We know that, in the discharge of a jar, a charge of elec- 
tricity is transferred from a point at a higher potential to one 
at a lower. It is reasonable, therefore, to suppose the phe- 
nomena under consideration to be also due, in some way, to 
the transfer of electricity from a higher to a lower potential. 
Since these phenomena continue without interruption while 
the circuit is joined up, it is necessary to assume that the vol- 
taic battery maintains a permanent difference of potential. 
This power of maintaining a difference of potential is ascribed 
to an electromotive force existing in the circuit. 

In an actual circuit containing a voltaic battery, if two 
points on the circuit outside the battery be tested by an elec- 
trometer, a difference of potential between them will be found. 
If the circuit be broken between the two points considered, 
the difference of potential between them becomes greater. 
This maximum difference of potential is the sum of finite 
differences of potential supposed to be due to molecular inter- 
actions at the surfaces of contact of different substances in the 
circuit, and is the measure of the electromotive force. An 
electromotive force may exist in a circuit in which there are 
no differences of potential. These cases will be considered 
later. It is sufficient for the present to consider two points 
between which a difference of potential is maintained, and 
which are connected by conductors of any kind whatever. 

The dimensions of electromotive force in the electrostatic 
system are those of difference of potential, or[ A] = W7*L?T7-". 


202] THE ELECTRICAL CURRENT. Be, 


202. Electrostatic Unit of Current.—Let us denote the 
potentials at the two points I and 2 in the circuit by V, and 
V,,and let V, be greater than V,; then if, in the time ¢, a quan- 
tity of electricity equal to Q passes through a conductor join- 
ing those points from potential V, to potential V,, the amount 
of work done by it is Q(V, — V,). 

If the conductor be a single homogeneous metal or some 
analogous substance, and no motion of the conductor or of 
any external magnetic body take place, the whole work done is 
expended in heating the conductor. If wesuppose the transfer 
to be such that equal quantities of heat are developed in equal 
times, we may represent the heat produced in the time ¢ by 
fit, if H represent the heat developed in one unit of time. If 
all the quantities considered are expressed in terms of the same 
fundamental units, we have 


OV,-V, =H, o H=2y,—7). 


The transfer of electricity in the circuit is called the electrical 
Q . 
current, and the rate of transfer or T is called the current 


strength, or often simply the current. The current, as here de- 
fined, is independent of the nature of the conductor, and is the 
same for all parts of the circuit. This fact was experimentally 
proved by Faraday. Employing this quantity /, we have the 
fundamental equation 


H=KV,—V,). (85) 


If heat and difference of potential be measured in absolute 
units, this equation enables us to determine the absolute mwmz¢ 
of current. The system of units here used is the electrostatic 
system. The dimensions of current strength in the electro- 


270 ELEMENTARY Pid Yate: [203 


static system are obtained from the equation above. We have 
OQ f : 
Pa =] = M*L?T-?, the dimensions of current. 


203. Ohm’s Law.—In § 187 it was remarked that a body 
is distinguished as a good ora poor conductor by the rate at 
which it will equalize the potentials of two electrified conduc- 
tors, if it be used to connect them. Manifestly this property 
of the substances forming a circuit, of conducting electricity 
rapidly or otherwise, will influence the strength of the current 
in the circuit. It was shown on- theoretical considerations, in 
1827, by Ohm of Berlin, that ina homogeneous conductor which 
is kept constant, the current varies directly with the difference 
of potential between the terminals. If A represent a factor, 
constant for each conductor, Oxm’s law is expressed in its sim- 
plest form by 


IR=V.—V, (86) 


The quantity £ is called the vesestance of the conductor. Ifthe 
difference of potential be maintained constant, and the conduc- 
tor be altered. in any way that does not introduce an internal 
electromotive force, the current will vary with the changes in 
the conductor, and there will be a different value of AR with 
each change in the conductor. The quantity & is therefore a 
function of the nature and matérials of the conductor, and 
does not depend on the current or the difference of potential 
between the ends of the conductor. Since itis the ratio of the 
current to the difference of potential, and since we know these 
quantities in electrostatic units, we can measure & in electro- 
static units. From the dimensions of / and (V, — V,) we may 
obtain the dimensions of &. They are in electrostatic units 


Vi — 


fs) = a 
Fin) eee 


[R] = |, 


203] THE ELECTRICAL CURRENT. 277 


To generalize Ohm's law for the whole circuit, let us con- 
sider a special circuit which may serve as a 
type. It shall consist of a voltaic cell contain-- on 
ing acidulated water, in which are immersed a 
zine and a platinum plate, joined together by 
a platinum wire outside the liquid (Fig. 62). 
Consider a point in the liquid just outside the 
zinc; if the potential of a point near it, just 
inside the zinc, be Vz, then the potential at the 
point considered is Vz+ 2/L, if Z2/L represent the sudden 
change in potential across the surface of separation. The 
potential at a point in the liquid just outside the platinum is 
Vz, and by the elementary form of Ohm’s law already con- 
sidered we have 


Pt 


Fic. 62. 


Se as 


yi R, 


In the same way the current in the platinum and platinum 
wire is expressed by 


PMT PRL 
Fd Aaa ot SANS SR) 
Rp 


and in the zinc by 


Vet P/Z — Vz 
f=. 
he 


Now these currents are all equal, for there is no accumulation 
of electricity anywhere in the circuit. Hence 


By 20s LEAS AT, Mn Aes Me 
i Rt i Rp 
Vp + P/Z — Vz 

ee 


fs 


— 
— 


278 ELEMENTARY PHYSICS. [204. 


or 


7 SEA LIPTPIE 
ERE AS Reem 


But the numerator is the sum of all the differences of potential 
in the circuit taken in one direction, or the measure of the 
electromotive force, and the denominator is the total resistance 
of the circuit. It may then be stated more generally as Ohm's 
Jaw that in any circuit the current equals the electromotive 
force divided by the resistance, or 


E 


204. Specific Conductivity and Specific Resistance. — 
If two points be kept at a constant difference of potential, and 
joined by a homogeneous conductor of uniform cross-section, 
it is found that the current in the conductor is directly propor- 
tional to its cross-section and inversely as its leneth. The cur- 
rent also depends upon the nature of the conductor. If con- 
ductors of similar dimensions, but of different materials, are 
used, the current in each is proportional to a quantity called 
the specific conductivity of the,material. The numerical value 
of the current set up in a conducting cube, with edges of unit 
length, by unit difference of potential between two opposite 
faces, is the measure of the conductivity of the material of the 
cube. The reciprocal of this number is the specific resistance 
of the material. If o represent the specific resistance of the 
conducting material, S the cross-section and / the length of a 
portion of the conductor of uniform cross-section between two 
points at potentials V, and V,, Ohm’s law for this special case 
can be presented in the formula 


Ff Nee Vi Vs) 


Zz (88) 


- 


206] THE ELECTRICAL CURRENT. 279 


The specific resistance is not perfectly constant for any one 
material, but varies with the temperature. In metals the spe- 
cific resistance increases with rise in temperature; in liquids 
and in carbon it diminishes with rise in temperature. Upon 
this fact of change of resistance with temperature is based a 
very delicate instrument, called by Langley, its inventor, the 
bolometer, for the measurement of the intensity of radiant 
energy. 

205. Joule’s Law.—If we modify the equation H = 
I(V, — V,) by the help of Ohm’s law, we obtain 


H= TPR. (80) 


The heat developed in a homogeneous portion of any cir- 
cuit is equal to the square of the current in the circuit multi- 
plied by the resistance of that portion. This relation was first 
experimentally proved by Joule in 1841, and is known after 
his name as Youle’s daw. It holds true for any homogeneous 
circuit or for all parts of a circuit which are homogeneous. 
The heat which is sometimes evolved by chemical action, or by 
the Peltier effect, occurs at non-homogeneous portions of the 
circuit. 

206. Counter Electromotive Force in the Circuit.—In 
many cases the work done by the current does not appear 
wholly as heat developed in accordance with Joule’s law. 

Besides the production of heat throughout the circuit, work 
may be done during the passage of the current, in the decom- 
position of chemical compounds, in producing movements of 
magnetic bodies or other circuits in which currents are passing, 
or in heating junctions of dissimilar substances. 

Before discussing these cases separately we will connect 
them all by a general law, which will at the same time present 
the various methods by which currents can be maintained. 
They differ from the simple case in which the work done ap- 


280 ELEMENTARY PHYSICS. [206 


pears wholly as heat throughout the circuit, in that the work 
done appears partly as energy available to generate currents in 
the circuit. To show this we will use the method given by 
Helmholtz and by Thomson. The total energy expended in the 
circuit in the time 7, which is such that, during it, the current 
is constant, is /A¢. It appears partly as heat, which equals 
I’?Rt by Joule’s law, and partly as other work, which in every 
case is proportional to /, and can be set equal to /A, where A 
is a factor which varies with the particular work done. Then 
we have /Et = J/’?Rt + JA, whence 


) hwo mas re (90) 


A 
It is evident from the equation that & — > is an electromo- 


tive force, and that the original electromotive force of the cir- 
cuit has been modified by the fact of work having been done 
by the current. In other words, the performance of the work 
LA in the time ¢ by the circuit has set up a counter electromo- 


A é 
tive force oy The separated constituents of the chemical com- ' 


pound, the moved magnet, the heated junction, are all sources 
of electromotive force which oppose that of the original circuit. 
If then, in a circuit containing no impressed electromotive 
force, or in which & = 0, there be brought an arrangement of 
uncombined chemical substances which are capable of com- 
bination, or if in its presence a magnet or closed current be 
moved, or if a junction of two dissimilar parts of the circuit be 


; t A 
heated, there will be set up an electromotive force - and a 


A 


; , 
Current ./ f= RP Any of these methods may then be used as 


206] THE ELECTRICAL CURRENT. 281 


the means of generating a current. The first gives the ordi- 
nary battery currents of Volta, the second the induced cur- 


rents discovered by Faraday, and the third the thermo-electric 
currents of Seebeck. 


CHAP ane Rais 
CHEMICAL RELATIONS OF THE CURRENT. 


207. Electrolysis.—It has been already mentioned that, in 
certain cases, the existence of an electrical current in a circuit 
is accompanied by the decomposition into their constituents of 
chemical compounds forming part of the circuit. This process, 
called electrolysis, must now be considered more fully. It is 
one of those treated generally in § 206, in which work other 
than heating the circuit is done by the current. That work is 
done by the decomposition of a body the constituents of 
which, if left to themselves, tend to recombine, is evident from 
the fact that, if they be allowed to recombine, the combina- 
tion is always attended with the evolution of heat or the ap- 
pearance of some other form of energy. The amount of heat 
developed, or the energy gained, is, of course, the measure of 
the energy lost by combination or necessary to decomposi- 
tion. 

A free motion of the molecules of a body, associated with 
close contiguity, seems to be necessary in order that it may be 
decomposed by the current. Only liquids, and solids in solu- 
tion or fused, have been electrolysed. Bodies which can be 
decomposed were called by Faraday, to whom the nomencla- 
ture of this subject is due, electrolytes. ‘The current is usually 
introduced into the electrolyte by solid terminals called elec- 
trodes. The one at the higher potential is called the positive 
electrode, or anode; the other, the negative electrode, or cathode. 
The two constituents into which the electrolyte is decom- 
posed are called zovs. One of them appears at the anode and 


207] CHUMICALIRE LATIONS OF THE CORRENT. 283 


is called the azzon, the other at the cathode and is called the 
cation. 

For the sake of clearness we will describe some typical 
cases of electrolysis. The original observation of the evolution 
of gas when the current was passed through a drop of water, 
made by Nicholson and Carlisle, was soon modified by Carlisle 
in a way which is still generally in use. Two platinum elec- 
trodes are immersed in water slightly acidulated with sulphuric 
acid, and tubes are arranged above them so that the gases 
evolved can be collected separately. When the current is pass- 
ing, bubbles of gas appear on the electrodes. When they are 
collected and examined, the gas which appears at the anode is 
found to be oxygen, and that which appears at the cathode to 
be hydrogen. The quantities evolved are in the proportion to 
form water. This appears to be a simple decomposition of 
water into its constituents, but it is probable that the acid in 
the water is first decomposed, and that the constituents of 
water are evolved by a secondary chemical reaction. 

An experiment performed by Davy, by which he dis- 
covered the elements potassium and sodium, is a good 
example of simple electrolysis. He fused caustic potash 
in a platinum dish, which was made the anode, and immersed 
in the fused mass a platinum wire as cathode. Oxygen was 
then evolved at the anode, and the metal potassium was de- 
posited on the cathode. ‘This is the type of a large series of 
decompositions. 

If, in a solution of zinc sulphate, a plate of copper be made 
the anode and a plate of zinc the cathode, there will be zinc 
deposited on the cathode and copper taken from the anode, 
so that, after the process has continued for a time, the solution 
will contain a quantity of cupric sulphate. This is a case simi- 
lar to the electrolysis of acidulated water, in which the simple 
decomposition of the electrolyte is modified by secondary 
chemical reaction. 


284 ELEMENTARY PHYSICS. [208 


—— 


If two copper electrodes be immersed in a solution of cu- 


pric sulphate, copper will be removed from the anode and de. 


posited on the cathode, without any important change occur- 
ring in the character or concentration of the electrolyte. This 
is an example of the special case in which the secondary reac- 
tions in the electrolyte exactly balance the work done by the 
current in decomposition, so that on the whole no chemical 
work is done. 

208. Faraday’s Laws.—The researches of Faraday in elec- 
trolysis developed two laws, which are of great importance in 
the theory of chemistry as well as in electricity. 

(1) The amount of an electrolyte decomposed is directly pro- 
portional to the quantity of electricity which passes through 
it; or, the rate at which a body is electrolysed is proportional 
to the current strength. 

(2) If the same current be passed through different electro. 
lytes, the quantity of each ion evolved is proportional to its 
chemical equivalent. 

If we define an electro-chemical equivalent as the quantity of 
any ion which is evolved by unit current in unit time, then 
the two laws may be summed up by saying: 

The number of electro-chemical equivalents evolved in a 
given time by the passage of any current through any electro- 
lyte is equal to the number of units of electricity which pass 
through the electrolyte in the given time. 

The electro-chemical equivalents of different ions are pro- 
portional to their chemical equivalents. Thus, if zinc sulphate, 
cupric sulphate, and argentic chloride be electrolysed by the 
same current, zinc is deposited on the cathode in the first case, 
copper in the second, and silver in the third: The amounts 
by weight deposited are in proportion to the chemical equiva- 
lents, 32.6 parts of zinc, 31.7 parts of copper, and 108 parts of 
silver. 

209. The Voltameter.—These laws were used by Faraday 


210| CHEMICAL RELATIONS OF THE CURRENT. 235, 


to establish a method of measuring current by reference to an 
arbitrary standard. The method employs a vessel containing. 
an electrolyte in which suitable electrodes are immersed, so 
arranged that the products of electrolysis, if gaseous, can be 
collected and measured or, if solid, can be weighed. This ar- 
rangement is called a voltameter. If the current strength be 
desired, the current must be kept constant in the voltameter 
by suitable variation of the resistance in the circuit during the 
time in which electrolysis is going on. 

Two forms of voltameter are in frequent use. 

In the first form there is, on the whole, no chemical work 
done in the electrolytic process. The system consisting of two 
copper electrodes and cupric sulphate as the electrolyte is an 
example of such a voltameter. The weight of the copper de- 
posited on the cathode measures the current. 

The second form depends for its indications on the evolu- 
tion of gas, the volume of which is measured. The water vol- 
tameter is a type, and is the form especially used. The gases 
evolved are either collected together, or the hydrogen alone is 
collected.” The latter is p.eferable, because oxygen is more 
easily absorbed by water than hydrogen and an error is thus 
introduced when the oxygen is measured. 

210. Measure of the Counter Electromotive Baee of 
Decomposition.—In the general formula developed in § 206 
the quantity /4 represents the energy expended in the circuit 
which does not appear as heat developed in accordance with 
Joule’s law. In the present case it is the energy expended 
during electrolysis in decomposing chemical compounds and 
in doing mechanical work. In many cases the mechanical work 
done is not appreciable; but when a liquid like water is decom- 
posed into its constituent gases, work is done by the expan- 
sion of the gases from their volume as water to their volume as 
gases. Let e represent the electro-chemical equivalent of one 
of the ions, and 6 the heat evolved by the combination of a 


286 ELEMENTARY PHYSICS. “2u7 


unit mass of this ion with an equivalent mass of the other ion, 
in which is included the heat equivalent of the mechanical work 
done if the state of aggregation change. Then / will represent 
the number of electro-chemical equivalents evolved, and /e@ will 
represent the energy expended, which appears as chemical sepa- 
ration and mechanical work. This is equal to JA; whence 
A=e06. All these quantities are measured in absolute units. 
The quantity ef represents the energy required to separate the 
quantity ¢ of the ion considered from the equivalent quantity 
of the other ion, and to bring both constituents to their normal 
condition. 

If the electrolytic process go on uniformly for a time 7#, so 
that equal quantities of the ion considered are evolved in equal 


A € A 
times, we have ie Now, lh represents the counter elec- 


tromotive force set up in the circuit by electrolysis. Hence 
the electromotive force set up in the electrolytic process may 
be measured in terms of heat units; or, since these heat units 
are measures of chemical affinity, the same relation gives a 
measure of chemical affinity in terms of electromotive force. 

It often is the case that the two ions which appear at the 
electrodes are not capable of direct recombination, as has been 
tacitly assumed in ine definition of 6. A series of chemical 
exchanges is always possible, however, which will restore the 
ions as constituents of the electrolyte, and the total heat evolved 
for a unit mass of one ion during the process is the quantity @. 

The theory here presented is abundantly verified by the ex- 
periments of Joule. Favre and Silbermann, Wright and others. 

211. Positive and Negative Ions.—Experiment shows 
that certain of thé bodies which act as ions usually appear at 
the cathode, and certain others at the anode. The former are 
called electro-positive elements; the latter, electro-negative ele- 
ments. Faraday divided all the ions into these two classes, 
aud thought that every compound capable of electrolysis was 


212] CHEMICAL RELATIONS OF THE CURRENT, 287 


made up of one electro-positive and one electro-negative ion. 
But the distinction is not absolute. Some ions are electro- 
positive in one combination and electro-negative in another. 
Berzelius made an attempt to arrange the ions in a series, such 
that any one ion should be electro-positive to all those above 
it and electro-negative to all those below it. It is questionable 
whether a rigorous arrangement of the ions is at the present 
time possible. } 

212. Theory of Electrolysis.—When any attempt is made 
to explain the behavior of the ions in the process of electroly- 
sis, grave difficulties are met with at once. The foundation 
of all the present theories is found in the theory published by 
Grotthus in 1805. He considers the constituent ions of a 
molecule as oppositely electrified to an equal amount. When 
the current passes, owing to the electrical attractions of the 
electrodes, the molecules arrange themselves in lines with their 
similar ends in one direction, and then break up. The electro- 
negative ion of one molecule moves toward the positive elec- 
trode and meets the electro-positive ion of the neighboring 
molecule, with which it momentarily unites. At the ends of 
the line an electro-negative ion with its charge is frecd at the 
anode, and an electro-positive ion with its charge at the 
cathode. This process is repeated indefinitely so long as the 
current passes. 

Faraday modified this view, in that he ascribed the arrange- 
ment or polarization of the molecules, and their disruption, to 
the stress in the medium which was the cardinal point in his 
electrical theories. Otherwise he held closely to Grotthus’ 
theory. He showed that the state of polarization existed in- 
the electrolyte by means of fine silk threads immersed in it. 
These arranged themselves along the lines of electrical stress. 

Other phenomena, however, show that Grotthus’ hypothesis 
can only be treated as a rough mechanical illustration of the 
main facts. 


288 ELEMENTARY PHYSICS. [212 


Joule showed that during electrolysis there is a development 
of heat at the electrodes, in certain cases, which is not accounted 
for by the elementary theory above given. It must depend 
upon a more complicated process of electrolysis than the one 
we have described. 

The results of researches on the so-called wandering of the 
ions are also at variance with Grotthus’ theory. If the electro- 
lysis of a copper salt, in a cell with a copper anode at the bot- 
tom, be examined, it will be found that the solution becomes 
more concentrated about the anode and more dilute about the 
cathode. These changes can be detected by the color of the 
parts of the solution, and substantiated by chemical.analysis. 
If this result be explained by Grotthus’ theory, the explanation 
furnishes at the same time a numerical relation between the 
ions which have wandered to their respective regions in the 
electrolyte which is not in accord with experiment. 

Another peculiar phenomenon, known as electrical endos- 
mose, may also be mentioned in this connection. It is found 
that, if the electrolyte be divided into two portions by a porous 
diaphragm, there is a transfer of the electrolyte toward the 
cathode, so that it stands at a higher level on the side of the 
diaphragm nearer the cathode than on the other. This fact 
was discovered by Reuss in 1807, and has been investigated 
by Wiedemann and Quincke. They found that the amount 
of the electrolyte transferred is proportional to the current 
strength, and independent of the extent of surface or the thick- 
ness of the diaphragm. Quincke has also demonstrated a flow 
of the electrolyte toward the cathode in a narrow tube, without 
the intervention of a diaphragm. Those electrolytes which are 
the poorest conductors show the phenomenon the best. In a 
very few cases the motion is towardsthe anode. The material 
of which the tube is composed influences the direction of flow. 
It has also been shown that solid particles move in the electro- 
lyte, usually towards the anode. 


212] CHEMICAL RELATIONS OF THE CURRENT. 289 


To explain these phenomena, Quincke has brought forward 
_a theory of electrolysis which is widely different from Grotthus’ 
simple hypothesis, but is too complicated for presentation here. 

It is an objection against Grotthus’ theory, and indeed 
against Thomson’s method given in § 210 of connecting chemi- 
eal affinity and electromotive force, that, on those theories, it 
would require an electromotive force in the circuit greater 


A ; 
than oy the counter electromotive force in the electrolytic 


cell, to set up a current, and that the current would begin sud- 
denly, with a finite value, after this electromotive force was 
reached. On the contrary, experiments show that the smallest 
electromotive force will set up a current in an electrolyte and 
even maintain one constantly, though the current strength may 
be extremely small. 

This is explained by Clausius by the help of the theory of 
the constitution of liquids which is now generally adopted. He 
conceives the molecules of the electrolyte to be moving about 
with different velocities. He thinks that occasionally the at- 
traction between two opposite ions of two neighboring mole- 
cules may become greater than that between the constituents 
of the molecules. In that case the molecules are broken up, 
the two attracting ions combine to form a new molecule, and 
two opposite ions are set free. These may at once combine to 
form another new molecule, or they may wander through ‘the 
mass until they meet with other ions, with which they can 
unite to again form molecules. He thinks that the electro- 
motive force in the circuit, while not great enough to effect a 
decomposition of the electrolyte, may yet be sufficient to deter- 
mine the direction of motion of these unpaired ions, so that 
they move, on the whole, towards their respective electrodes. 
Every theory of electrolysis assumes that the transfer of elec- 
tricity is, in some way, connected with the transfer of the ions; 
hence on Clausius’ theory there will be a current and an evolu- 

1g 


290 ELEMENTARY FH YsiCs. [213 


tion of the ions with any electromotive force in the circuit, 
however low. This current would at once cease if the ions 
were to collect on the electrodes, and set up a permanent 
counter electromotive force; but the same reasoning as has 
¢just been used will show that the liberated ions, if not formed 
ia such quantities as to collect and pass out of the liquid as 
in true electrolysis, will wander back into the liquid again. On 
this theory the number of free ions of either kind ought to be 
greater near the electrode to which they tend to move. 

While Clausius’ theory fully accounts for the behavior of 
the ions, it does not explain their relations to the electrical 
current. No satisfactory theory of the relations of electricity 
to the molecules of matter has as yet been given. 

213. Voltaic Cells.—From the discussion given in § 206 it 
is obvious that, if an arrangement be made, in a circuit, of sub- 
stances capable of uniting chemically and such as would result 
from electrolysis, there will result an electromotive force in 
such a sense as to oppose the current which would effect the 
electrolysis. If, then, the electrodes of an electrolytic cell in 
which this electromotive force exists be joined by a wire, a 
current will be set up through the wire in the opposite direc- 
tion to the one which would continue the electrolysis, and the 
ions at the electrodes will recombine to form the electrolyte. 
There is thus formed an independent source of current, the 
voltaic cell. The electrode in connection with the electro-nega- 
tive ion is called the posztzve pole, and that in connection with 
the electro-positive ion the zegatzve pole. 

Thus, if after the electrolysis of water in a voltameter, in 
which the gases are collected separately in tubes over platinum 
electrodes, the electrodes be joined by a wire, a current will be 
set up in it, and the gases will gradually, and at last totally, 
disappear, and the current will cease. The current which de- 
composes the water is conventionally said to flow through the 
liquid from the anode to the cathode, from the electrode above 


213] CHEMICAL RELATIONS OF THE, CURRENT, 291 


which oxygen is collected to the electrode above which hydro- 
gen is collected. The current existing during the recombina- 
tion of the gases flows through the liquid from the hydrogen 
electrode to the oxygen electrode, or outside the liquid from 
the positive to the negative pole. Such an arrangement as is 
here described was devised by Grove, and is called the Grove’s 
gas battery. 

A combination known as Smee’s cell consists of a plate of 
zinc and one of platinum, immersed in dilute sulphuric acid. 
It is such acell as would be formed by the complete electrolysis 
of a solution of zinc sulphate, if the zinc plate were made the 
cathode. When the zinc and platinum plates are joined bya 
wire, a current is set up from the platinum to the zinc outside 
the liquid, and the zinc combines with the acid to form zinc ° 
sulphate. The hydrogen thus liberated appears at the platinum 
plate, where, since the oxygen which was the electro-negative 
ion of the hypothetical electrolysis by which the cell was 
formed does not exist there ready to combine with it, it col- 
lects in bubbles and passes up through the liquid. The pres- 
ence of this hydrogen at once lowers the current from the cell, 
for it sets up a counter electromotive force, and also dimin- 
ishes the surface of the platinum plate in contact with the 
liquid, and thus increases the resistance of the cell. It may be 
partially removed by mechanical movements of the plate or by 
roughening its surface. The counter electromotive force is 
called the electromotive force of polarization. It occurs soon 
after the circuit is joined up in all cells in which only a single 
liquid is used, and very much diminishes the currents which are 
at first produced. 

Advantage is taken of secondary chemical reactions to avoid 
this electromotive force of polarization. ‘The best example, 
and a cell which is of great practical value for its cheapness, 
durability, and constancy, is the Danzell’s cell. Two liquids 
are used, solutions of cupric sulphate and zinc sulphate. They 


292 BLEMENTA RB) PTY St Cs: [ors 


are best separated from one another by a porous porcelain 
diaphragm. A plate of copper is immersed in the cupric sul- 
phate, and a plate of zinc in the zinc sulphate. The copper is. 
the positive pole, the zinc the negative pole. When the circuit. 
is made and the current passes, zinc is dissolved, the quantity 
of zinc sulphate increases and that of the cupric sulphate de- 
creases, and copper is deposited on the copper plate. To pre- 
vent the destruction of the cell by the consumption of the 
cupric sulphate, crystals of the salt are placed in the solution. 
The electromotive force of this cell is evidently due to the 
loss of energy in the substitution of zinc for copper in the 
solution of cupric sulphate. It may be calculated by the for- 
mula of §210. The experiments of Kohlrausch give for zine 
in C. G. S. units, e = 0.003411, where the system of units em- 
ployed is the electromagnetic (§ 218). Favre and Silbermann 
give for 6, in the chemical process here involved, 714 gram- 
degrees or lesser calories. The mechanical equivalent of one 
gram-degree is 41,595,000. Hence we obtain for the electro- 
motive force of a Daniell’s cell in C. G. S. electromagnetic 
units the value 1.013-10°. The value as found by direct ex- 
periment is about 1.1-10° in C. G. S. electromagnetic units. 

There are many other forms of cell, which are all valuable 
for certain purposes. One of the best known is the Grove's 
cell. It has for positive pole a platinum plate; immersed in 
strong nitric acid, and for negative pole a zinc plate, immersed 
in dilute sulphuric acid. The two liquids are separated by a 
porous porcelain diaphragm. When the current passes, the 
zinc is dissolved. The hydrogen freed is oxidized by the nitric 
acid, which is gradually broken up into other compounds. 
The electromotive force of the Grove’s cell is very high, being 
about 1.88-10° C. G.S. electromagnetic units. 

The secondary cell of Planté is an example of a cell made 
directly by electrolysis, as has been assumed in the preliminary 
discussion. ‘The electrodes are both lead plates, and the elec- 


214] GHEMICAT, RELATIONS OF (JHE CURRENT. 293 


trolyte dilute sulphuric acid. When a current is passed 
through the cell, the oxygen evolved on the anode combines 
with the lead to form peroxide of lead, which coats the surface 
of the electrode. When the cell is inserted in a circuit, a cur- 
rent is set up, the peroxide is reduced to a lower oxide, and 
the metallic lead of the other plate is oxidized. 

The Latimer-Clarke standard cell is of great value as a 
standard of electromotive force. As it polarizes at once if a 
current pass through it, it should never be joined up ina 
closed circuit. The positive pole consists of pure mercury, 
which is covered by a paste made by boiling mercurous sul- 
phate in a saturated solution of zinc sulphate. The negative 
pole consists of pure zinc resting on the paste. Contact with 
the mercury is made by means of a platinum wire. As no 
gases are generated, this cell may be hermetically sealed against 
atmospheric influences. According to the measurements of 
Rayleigh, the electromotive force of this cell is very constantly 
1.435-10° C.G. S. electromagnetic units at 15° Cent. 

214. Theories of the Electromotive Force of the 
Voltaic Cell.—The plan followed in the preceding discussions 
has rendered it unnecessary for us to adopt any theory to ex- 
plain the cause of the electromotive force of the voltaic cell. 
The different theories which have been advanced may be 
classed under one of two general theories, the contact theory 
and the chemical theory. On the contact theory, as advanced 
by Volta and supported by Thomson and others, the difference 
of potential which exists between two heterogeneous substances 
in contact is due to molecular interactions across the surface 
of contact, or, as it is commonly stated, is due merely to the 
contact. The chemical theory, as advocated by Faraday and 
Schoénbein, holds that the difference of potential considered 
cannot arise unless chemical action or a tendency to chemical 
action exist at the surface of contact. 

Numerous experiments have shown that the sum of all the 


294 ELEMENTARY PHYSICS. [215, 


differences of potential at the surfaces of contact of the various 
substances making up any voltaic cell is equal to the electro- 
motive force of that cell. This is true even when the cell is 
formed solely of liquid elements. On the contact theory this 
electromotive force is due merely to the several contacts, while 
the chemical actions of the cell begin only when the circuit is 
made, and supply the energy for the maintenance of the cur- 
rent. The quantity of heat produced at a junction of dis- 
similar substances by the passage of a current (§ 233) is such 
as to show, however, that the differences of potential thus 
measured are not the true differences of potential due to the 
contact of the substances tested, but must depend in part 
upon the action of the air or other medium by which these 
substances are surrounded. ‘The supporters of the chemical 
theory point to this fact as evidence that the chemical action 
of the medium is concerned in the production of the difference 
of potential observed. 

On either theory it is clear that the energy maintaining the 
current must have its origin in the chemical actions which go 
on in the voltaic cell. 

215. Capillary Electrometer.—It has been stated that a 
‘difference of potential exists between a metal and a fluid elec- 
trolyte in contact with it. There will then exist on the sur- 
faces of the metal and the electrolyte in contact with it such 
an electrical distribution as exists in a charged condenser of 
which the plates are very near together. 

One arrangement by which the effects due to this distribu- 
tion may be observed was devised by Lippmann. It consists 
of a vertical glass tube, drawn out at its lower end in a capil- 
lary tube. The capillary tube dips into dilute sulphuric acid, — 
which rests on mercury in the bottom of the vessel containing 
it. Mercury is poured into the vertical tube until its pressure 
is such that the capillary portion of the tube is nearly filled 
with it. When the mercury in the vessel is joined with the 


215] CHEMICAL RELATIONS OF THE CURRENT. 295 


positive pole of a voltaic cell, and that in the tube with the 
negative pole, the meniscus in the capillary tube moves up- 
ward, in the sense in which it would move if its surface tension 
were increased. This movement may best be explained by the 
help of the theory of electrolysis given by Clausius ($212). So 
long as there exists an electromotive force in the circuit, posi- 
tive and negative ions will be released on their respective elec- 
trodes. If we assume that they are associated with the trans- 
fer of electricity in the circuit in such a way that it is trans- 
ferred from them to the electrodes, such a movement of the 
ions would give rise to a modification of the distribution on 
the surfaces of contact. In the case now under consideration 
the charge on the meniscus is in part neutralized by the charge 
transferred with one of the ions. The true surface tension of 
the surface of separation between the mercury and the liquid 
is, on this theory, lessened by the presence of the electrical 
charge on the surfaces of contact, owing to the interaction of 
the parts of the charge ina manner similar to that described in 
§i1gt. If, therefore, any diminution of this charge occur, a 
seeming increase of the surface tension will be observed. On 
this theory the true surface tension of the surface of separation 
is the value observed when the mercury and liquid are at the 
same potentials, and this value is a maximum. The experi- 
ments of A. Kénig and Helmholtz show that such a maximum 
value exists in a manner consistent with the theory. 

The arrangement described can manifestly be used to pro- 
duce the effects just discussed only when the electromotive 
force introduced into the circuit is less than that required to 
cause active decomposition of the electrolyte. If any suitable 
electromotive force be introduced into the circuit, the theory 
here given assumes that the transfer of the ions goes on until 
the differences of potential on the surfaces of contact are such 
as to counterbalance the introduced electromotive force. The 
mercury column then comes to rest. 


296 ELEMENTARY PHYSICS. [215 


Lippmann constructed an apparatus similar to the one de- 
scribed, with the addition of an arrangement by which pressure 
can be applied to force the end of the mercury column in the 
capillary tube back to the fixed position which it occupies 
when no electromotive force is introduced into the circuit. He 
found that when small electromotive forces were introduced, 
the pressures required to bring the end of the column back to 
the fixed position were proportional to the electromotive 
forces. He hence called this apparatus a capillary electrometer. 

Lippmann also found that if the area of the surface of 
separation between the mercury and the liquid in the capillary 
tube were altered by increasing the pressure and driving the 
mercury down the tube, a current was set up in a galvanome- 
ter inserted in the circuit, in a sense opposite to that which 
would change the area of the meniscus back to its original 
amount. 


CHCA PB Re V. 
MAGNETIC RELATIONS OF THE CURRENT. 


216. Biot’s Law.—Very soon after the discovery by Oer- 
sted of the fact that a magnet was acted upon by an electrical 
current brought near it, Biot and Savart instituted a series of 
experiments to determine the law of the force between a mag- 
net andacurrent. They suspended a short magnet by a silk 
fibre, and so modified the earth’s magnetic field near it, by 
means of magnets, that the suspended magnet pointed in any 
azimuth with equal freedom. A current was then passed 
through a long vertical wire near the magnet. It was observed 
that the magnet placed itself so that its poles were equally 
distant from the wire. The movement of the north pole and the 
direction of the current were related as the rotation and pro- 
pulsion in a right-handed screw. Then the magnet was set in 
oscillation, and the times of oscillation determined when the 
current was at different distances from the magnet, and when 
different currents were set up in the wire. From the first ob- 
servation it follows that the force exerted between a magnet 
pole and a current is normal to the plane passing through the 


current and the magnet pole. For, suppose the current rising 
vertically out of the paper at C (Fig. 63), and suppose that it 


298 ELEMENTARY PHYSICS. [216 


acts on the north pole of the magnet zs with a force repre- 
sented by za, making any angle @ with the line zC. It is as- 
sumed as probable that the force on a south pole placed at z 
would be oppositely directed to za. The angle which the force 
so, acting on the south pole s, makes with the line sC will 
then be w — ¢@. Now the magnet is in equilibrium, hence the 
moments of the components of these forces at right angles to 
the magnet must be equal. The components are respectively 
na sin (jf — @) and sb sin (bf — (wa — @)). The lever arms oz 
and os are equal, and it is assumed that, since the poles are at 
equal distances from the current, the forces wa and sé are 
equal; therefore sin ( — ¢) must equal sin (»b — (x — @)), 


1 
and this is true only when @=-. The lines of magnetic force 


about an infinite straight current are therefore circles, and the 
equipotential surfaces determined by these lines are planes 
passing through the current. 

From the times of oscillation observed, it was proved that 
the force exerted is proportional directly to the strength of 
the magnet pole and to the strength of the current, and in- 
versely to the distance between the pole and the current. Biot 
hence deduced a law for the action of each element of length 
of the current upon a magnet pole, which is expressed in the 
formula 


mt sin ads 


j= or (91) 


r 


In this formula m represents the strength of the magnet pole, 
z the current strength measured in electromagnetic units, ds 
the element of the current, 7 the distance between that ele- 
ment and the magnet pole, and @ the angle between 7 and ds. 
It is easy to show that the force exerted by a long straight 
current, observed by Biot to be inversely as the distance from 


216] MAGNETIC RELATIONS OF THE CURRENT. 299 


the current, is consistent with this law. For simplicity we will 
consider an infinitely long straight current. Let the magnet 


pole m be at the point P(Fig. 64). Let QF be the current 


x Sueree 


Fe b U 


Bic. 64. 


element ds, and PO the perpendicular distance between the 
pole and the current. Then Biot’s law gives for the force ex- 
mOR PO 
PRM PR 
limit, as QR becomes indefinitely small, the triangles ORS and 


erted by the element QA the expression —,; In the 


ep 
POR become similar. Hence QS equals Soe e and the ex- 
mt OS 


ae - lPabout.£, with Lobas 


radius, we draw the-arc OU, the elementary arc a in the limit 


pression for the force becomes ~ 


o. 
equals aes ae , and the projection 8 of the arc @ on the line 
aPO 2 
PU equals PR’ Using these values, the expression aes be- 


ae There will be a similar expression for the force 


due to any other element. The total force due to the whole 


1110 
comes —=- 


mut 
current will be equal to the constant factor =~ 


Po? multiplied by 


3CcO PLM LINE AY PESOS [217 


the sum of all the projections corresponding to &. This sum, 
for the infinite current, is manifestly 2PU = 2PO. Hence the 
oe or, it is inversely as the distance PO be- 
tween the pole and the current. 

217. Equivalence of a Closed Circuit and a Magnetic 
Shell.—The law of the force between a pole and a current, 
which has been stated, leads to the conclusion that a very small 
closed plane circuit, carrying a current, will act upon a magnet 
pole at a distance from it in the same way as a magnetic shell, 
of which the edge coincides with the contour of the circuit, and - 
the strength equals the strength of the current. To show this 
we will use a rectangular circuit with indefinitely small sides. 
We will place the origin (Fig. 65) at the centre of the rectan- 


total force is 


28 


- 
-_ 
a, 
on 
- 
- 
ad 
- 
- 
- 
- 
- 
- 
- 
an 
= 


rc 
! 
‘ 
| 
oe ee 
- ee ak Zz 
=e : 
. { 
Pitre est \ 
~~ yz 
Fic. 65 


gle, and draw the x axis perpendicular to the plane of the rect- 
angle, and the y and g axes parallel with its sides. For con- 
venience, we will call the length of the sides parallel to the » 
axis 2s, and of those parallel to the ¢ axis 2s’, 

Weassume that a current of strength z traverses the bound- 
ary of the rectangle in a direction related to the positive di- 
rection of the x axis, as the motions of rotation and propul- 
sion are related in a right-handed screw. 

If the magnet pole be at the point (vyz), the force onit due 


mr | MAGNETIC RELATIONS OF THE CURRENT. 30f 


to one side, 2s, is, as stated in Eq. (90), proportional to the 
length 2s, is inversely as the square of the distance 


ey acter) 3 


and is proportional to the sine of the angle between the line join- 
ing (xyz) and the element 2s and the direction of that element. 


This sine is expressed Seems: ae fom a in yr The total force 
m2 s(x te (eone S ; 

due to the element is then . This force is. 
tyes wi; 


at right angles to the plane passing through the direction of 

the element 2s and the perpendicular from (yz) on the direc- 

tion of that element. We shall investigate in turn the compo- 

nents along the three axes. That along the 4 axis is found by 
fi 


K=O, 


Ore —syF 


for the component along the + axis then becomes 


multiplying the total force by The expression 


2mis(g — s’) 
earn Ss) 


We will expand (¢ — s’)’ in the denominator, reject the term 
s’, remembering that the sides are indefinitely small, and write 
for brevity 77 + 7°+ 2° =7°. We then have this component 
expressed by —, Lad aE a 
(7? — 285')8 

components due to each of the other sides, with the difference 
that those due to opposite sides must have different signs. 
We call those positive which are directed along the positive 
direction of x. 

We will write the four components, and opposite them 
their expansions in ascending powers of s ors’, rejecting all 
terms containing the second or higher powers of s. 


Similar expressions hold for the 


302 ELEMENTARY PHYSICS. [217 


2mis(s — S’) j 
7 ( — 2888 = — 2mis(z — s')\(r-3 +.35'a7-5) § 


QUIS tS) ae nee oe 
Aamir + 2mis(e + s'\(r—3 — 38’er-5); 
ZULIS Yh) es =p a af 

NDApr mete ye a amis'(y — s\(r a 35y7r —5) ; 
2mis'( y+ s) 


ve = + 2mis'(y + s)\(r-3 — 357-5). 


(7" Pays) 


If we write out the sum of these expressions, rejecting all 
terms of the dimensions of s°*, we obtain as the component 
along the # axis of the force due to the whole circuit the ex- 


ae oe 
. ‘4 3y + 34 2 * . 
pression — 4miss'( 23 mma F The term in parenthesis 
2 2 2 2 2 
37h 38 sree ih oem 
tan be written \———- ye ana The factor 4ss’ is 


equal to a, the area of the rectangle. The force along the x 
axis is then finally 


i mia, = 2 (92) 


For the component along the y axis we have to consider 
only the forces due to the sides 2s’, for the other sides have no 
tendency to move the pole parallel with themselves. The com- 
ponents of these forces along y, that one being called positive 


which is in the positive direction of y, are + 27S aa 
y 2s 


and — 2mzs’ The sum of these components is 


vidion Mate 
(7? + 2ys)t 


A4miss’ a = mia? (92a) 


217] MAGNETIC RELATIONS OF THE CURRENT. 303 


Similarly the total component Bre the g axis is 


dks 
mia. (924) 


Now to compare these forces with those due to a magnetic 
shell of the indefinitely small area a and strength 7, we use the 
result of the discussion in § 182, that the potential of such a 
shell at any external point is 7@. In that discussion the con- 
vention was made that the positive face of the shell was turned 
toward the positive direction of the 4 axis. We then have o, 
the solid angle subtended by the shell as seen from the point 
P, equal to 


acos 6 AX x 


ea 
The potential at the point P is then 


Y= Sry THF 


To find the forces along the three axes we must find the rate 
of change of this potential with respect to space. To do this 
for the x axis, let x increase by a small increment 4%; then 
the potential will take a small increment JV. We will have 


cas eells diem eee h 
VT AV IU Ay yp pee? 


and as 4% becomes indefinitely small, 


at 4Ar 
VtAV= Ug rena Geert 


304 ELEMENTARY PHYSICS. [217 


Expanding this expression, rejecting all terms containing the 
second or higher powers of 4x, we obtain 


Vn jal ul ade a ae 


7 


From this we have further 


PI 32) 
Azia Waeee 


‘if Tr 


In the limit, as 4% becomes indefinitely small, this is the rate 
of change of potential along the + axis at the point (yz). 

The force along the x axis on a unit magnet pole at the 
point (xyz) is this rate of change of potential taken with the 
opposite sign. Hence the force on the magnet pole m at that 


point is — mjal — =). Similarly the forces along the y and 


2\ 


g axes can be found to be respectively mya and mya. 

If these expressions be compared with the expressions for 
the components of force arising from the action of the rect- 
angular current, they will be seen to be completely identical, 
provided that the unit of current be so selected that the 
factors z and 7 are equal. 

If the current in the circuit be reversed, the components of 
force due to it remain the same in amount but are opposite in 
direction. The direction of current in the circuit which will 
render its action completely identical with that of the mag- 
netic shell may be readily stated. Let us draw a line through 
the magnetic shell, tangent to the lines of force, from the 
negative to the positive face, and call its direction the positive 
direction of the lines of force. Then the current in the equiv- 


217} WAGNETICO RELATIONS OF (LHE CURRENT. 305 


alent circuit is such that its direction is related to the positive 
direction of the lines of force as the motions of rotation and 
propulsion are related in a right-handed screw. 

It may now be shown that a finite circuit of any form 
carrying a current z is equivalent to a magnetic shell of uni- 
form strength 7, the edge of which coincides with the circuit. 
For a finite circuit may be conceived to be made up of an 
assemblage of elementary circuits of the kind considered, 
lying contiguous to one another in the surface bounded by the 
contour of the circuit. Everywhere the currents of one of 
these elementary circuits is neutralized by the equal and in. 
finitely near currents in the opposite direction of the contigu- 
ous circuits, except at the boundary, where all the elementary 
currents are in the same direction and are equivalent to the 
current in the circuit. This reasoning will be plain at once 
irom Fig. 66. The forces due to 
such a current will then be equal to 
the forces due to a magnetic shell 
made up of elements which corre- 
spond to ihe elementary circuits. 
‘ihe systems of lines of force due to Fic. 66. 
the shell and the equivalent circuit will be precisely similar in 
form and distribution. ‘They will differ, however, in this, that 
the line of force joining two contiguous points on opposite 
faces of the shell will be interrupted by the shell, while in the 
case of the circuit it passes through the circuit as a continuous 
Jine enclosing the current. If a unit positive magnet pole were 
placed at a point on the positive face of a magnetic ‘shell, it 
would move along a line of force to a point infinitely near the 
one from which it started, but on the opposite or negative face 
of the shell, and during the movement it would do an amount 
of work expressed by 477. This same amount of work would 
be done upon it if it were brought back by any path to the 


point from which it started, so that the total work done in the 
20 


306 ELEMENTARY PHYSICS. (207 


closed path ts zero. If, on the other hand, the pole were mov- 
ing under the influence of the circuit equivalent to the mag- 
netic shell, it would move, as in the case of the shell, along the 
line of force from the positive to the negative face of the 
circuit, and in so doing would do work equal to 4zz. But 
from the fact that the line of force on which it is moving is 
continuous, and that the force in the field is everywhere finite, 
it would pass over the infinitesimal distance between the point 
on the negative face and the one on the positive face, from 
which it started, without doing any finite work. The system 
would then have returned to its original condition, and work 
equal to 47zz would have been done. This is expressed by 
saying that the potential of a closed current is mu/ltzply-valucd. 
The work done during any movement depends not only on the 
position of the initial and final points in the path, as in the 
case of the ordinary single-valued gravitational, electrical, and 
magnetic potentials, but also on the path traversed by the 
moving magnet pole. Every time the path encloses the cur- 
rent, work equal to 47z is done. The work done in moving by 
a path which does not enclose the current, from a point where 
the solid angle subtended by the circuit is @, to one where it 
is @, is, as in the case of the magnetic shell, equal to 2(@, — @). 
If the path further enclose the current z times, the work done 
is 472, so that the total work done, or the total difference of 
potential between the two points, is 


V, -V=uo, —w+4zn), (93) 


where z may have any value from o to infinity. 

The fact that the potential of a current is multiply-valued 
is well illustrated by any one of a series of experiments due to 
Faraday. If we imagine a wire frame forming three sides of a 
rectangle to be mounted on a support so as to turn freely about 
one of its sides as a vertical axis, while the free end of the 


218] WAGNETIC RELATIONS QF. THE CURRENT. 407 


opposite side dips in mercury contained in a circular trough of 
which the axis of rotation passes through the centre, and if we 
suppose a current to be sent through the axis and the frame, 
passing out through the mercury; then if a magnet be placed 
vertically with its centre on the level of the trough, and with 
either pole confronting the frame, the frame will rotate con- 
tinuously about the axis. “fey belwual are Trion f 7) 

Other arrangements are made by eich more cowiphicated 
rotations of circuits can be effected. If the circuit be fixed 
and the magnet movable, similar arrangements will give rise 
to motions of the magnet or to rotations about its own axis. 

218. Electromagnetic Unit of Current.—The relation 
which has been discussed between a circuit and the equivalent 
magnetic shell affords a means of defining a unit of current dif- 
ferent from that before defined in the electrostatic system. 
That current is defined as the wxz¢t current, which will set up 
the same magnetic field as that due to a magnetic shell of 
which the edge coincides with the circuit, and the strength is 
unity. 

This definition is equivalent to the following one, which is 
sometimes given. If the orce between a unit magnet pole 
and a current flowing in a plane circuit of .unit length, every 
part of which is at unit distance from the pole, be the unit 


force, then the current is the unit current. 

The equivalence of the two definitions may be shown as follows: Consider 
a circular plane magnetic shell of strength 7 set up normal to the « axis with its 
centre at the origin. We will determine the force at the point / on the ~ axis, 
along that axis, due to the shell. Designate the radius of the circular shell by 
&, and the angle contained between the w axis and a line drawn from the point 
p to any point on the boundary of the shell by @ Then 27k*(1 — cos ) rep- 
resents the area of the spherical calotte, of which the centre is the point /, and 
the edge the boundary of the shell. Hence 22(1 — cos @) is the solid angle 
subtended at the point # by the shell. We may express cos @ in other terms by 
ee and the potential V at the point g by V = 277 (: _ eae 
‘The rate of change of potential along the x axis measures the force required. 


308 ELEMENTARY PHYSICS. [218 


Let x change by asmall increment 4x. Then V will take an increment JV, 
and we have 


x+ 4x 
V+ AVi= 2707 ( rome ee 


If we expand this expression, we have in the limit, as 2x becomes in- 
definitely small, since we may neglect higher powers of 4x than the first, 


we ‘ a+ Ax 
Fae ie Gt a eapeererereere) 


cei B ( — (« + 4x) ee — CE 
as 277 ( NE nee pee nated Ba aa ee) 
(a? + FR?) (a? + FR) (x? + RB 
Hence we obtain 
atic Be 277 (Grae — ae, =— 277 eee 
Ax (a? ++ RF (2? + R?)t (x? + R)3 


2 
The force at the point # is therefore equal to 277 Cet When the 
z 


: . ae: : 277 
point / is at the origin, « equals 0, and the force is expressed by PR: If we 


adopt the first definition of unit current, and set 7 =7, the force on a magnet 
. 2mim 
R 


If we adopt the second definition of unit current, and use 
Biot’s formula for the action of a current on a magnet pole, 
the force due to a circular current, made up of current 

ms 


pole due to a circuit equivalent to the shell is 


elements of length s, upon a pole at its centre is = pe The 

sum of all the elements of the circle is 27. Hence the force 
j eae 21M 

on this definition is also R 


The unit based upon these definitions is called the electro. 
magnetic unit of current. It is fundamental in the construction 
of the electromagnetic system of units, in just the same way as 
the unit of quantity is fundamental in the electrostatic system. 


219] WMeGVe LORE LATIONS: OF THE CORRENT. 309 


In practice another wt of current, called the ampere, is used. 
It is equal to 107 C.G.S. electromagnetic units. The dimen- 
sions of the electromagnetic unit of current are those of the 
strength of a magnetic shell, or [2] = A*L*7 —'. 

219. Lines of Magnetic Force.—It is convenient, in 
much of the discussion of the action of currents, to use the 
notion of lines of force, and to measure the strength of field, 
as explained in § 21, by the number of lines of force. For 
example, we may conceive the field about a magnet pole to be 
filled with conical tubes of force, of an angular aperture which 
is very small, and equal for all the cones, but otherwise entirely 
arbitrary. It is commonly assumed that each one of these 
cones represents a line of force. Then the solid angle sub- 
tended by any magnetic shell in the field, which is measured 
by the number of the cones contained in that solid angle, can 
be replaced by the number of lines of force which the bound- 
ary of the shell encloses. 

If the magnet pole be free to move, it will move from a 
point of higher to a point of lower potential; that is, it will 
move in general to a point as near as possible to the negative 
face of the shell. If wemake the convention that a line of force 
passes through a shell in the positive direction when it passes 
from the negative to the positive face, we may describe this mo- 
tion as one of which the result is, that as many lines of force as 
possible pass through the shell in the positive direction. Ifthe 
magnet pole be fixed, and the shell free to move, it fellows, 
from the equality of action and reaction, that the shell will 
set itself so that as many lines of force as possible will pass 
through it in the positive direction. When the shell is not 
perfectly free to move, and in certain other special cases, it is 
sometimes convenient to use an equivalent statement, that the 
shell will move so that as few lines of force as possible pass 
through it in the negative direction. 

These last conclusions are independent of the particular 


310 ELEMENTARY PAV SICS. [220 


character of the magnetic field in which the shell is situated. 
It may then be stated generally, as a law governing the motions 
of magnetic shells or their equivalent electrical circuits in a 
magnetic field, that they tend to move so that as many lines 
of force as possible will pass through them in the positive 
direction. From the discussion in § 217 it may be seen that 
the positive direction of a line of force due to a current is re- 
lated to the direction of the current in the circuit as the direc- 
tions of propulsion and of rotation in a right-handed screw. 
To one looking at the negative face of a magnetic shell, the 
current in the equivalent circuit will travel with the hands of 
a watch. 

If a part only of the closed circuit be free to move, it may 
be considered by itself as a magnetic shell, and it will move in 
accordance with the same law. We can therefore use this law 
to investigate the movements of circuits or parts of circuits due 
to the magnetic field in which they are placed. 

220. Mutual Action of Two Currents.—In general, ae 
plane circuits, if they be free to move, will so place themselves 
that the lines of force from the positive face of one will pass 
through the other in the positive direction, or through its 
negative face. The currents in the two circuits will then have 
the same direction. If they be placed so that unlike faces are 
opposed, they will move towards one another; if so that similar 
faces are opposed, they will move away from one another. 
Since in the first case the currents are in the same direction, 
and in the second in opposite directions, the law may be stated 
in another form: that circuits carrying currents in the same 
direction attract one another; in opposite directions, repel one 
another. | 

Parts of the circuits, if movable, follow the same law. For 
example, consider a circuit in the form of a wire square, free 
to turn about a vertical line passing through the centres of two 
opposite sides. If nowa vertical wire, forming part of another 


221 | MAGNETIC RELATIONS OF THE CURRENT. 311 


circuit, be brought near one of the vertical sides of the square, 
that side will move towards the vertical wire, or away from it, 
according as the currents in the two wires are in the same or 
in opposite directions. It is clear that the maximum number 
of lines of force due to the fixed circuit pass through the mov- 
able circuit in the positive direction, when the two parallel 
portions carrying currents in the same: direction are as near 
one another as possible; and that as few lines of force as pos- 
sible pass through the movable circuit in the negative direction, 
when the two parallel portions carrying currents in opposite 
directions are as far from one another as possible. 

221. Ampére’s Law for the Mutual Action of Currents. 
—The laws of the action between electrical currents were first 
investigated by Ampere from a different point of view. From 
a series of ingenious experiments he deduced a law which ex- 
presses the action of a current element on any other current 
element. The action of any circuit on any other can be ob- 
tained from this law by summing the effects of all the elements. 
The complete deduction of the law from the experimental 
facts is too complicated to be given, but the experiments 
themselves are of great interest. 

Ampére’s method consisted in submitting a movable circuit 
. or part of a circuit carrying a current to the action of a fixed 
circuit, and so disposing the parts of the fixed 
circuit that the forces arising from different 
parts exactly annulled one another, so that the 
movable circuit did not move when the current 
in the fixed circuit was made or broken. In 
the first two of his experiments the movable 
circuit consisted of a wire frame of the form 
shown in Fig.67. The current passes into the 
frame by the points a and 4, upon which the 
frame is supported. It is evident that the two halves of the 
frame tend to face in opposite directions in the earth’s mag- 


Fic. 67. 


312 ELEMENTARY PHYSICS. [221 


netic field, so that there is no tendency of the frame as a whole 
to face in any one direction rather than any other. If a long 
straight wire be placed near to one of the extreme vertical sides 
of the frame and a current be sent through it, that side will 
move towards the wire if the currents in it and in the wire be 
in the same direction, and will move away from the wire if the 
currents be in opposite directions. 

If now this wire be doubled on itself,so that near the 
frame there are two equal currents occupying practically the 
same position, but in opposite directions, then no motion of the 
frame can be observed when a current is set up in the wire. 
This is Ampére’s frst case of equilibrium. It shows that the 
forces due to two currents, identical in strength and in posi- 
tion, but opposite in direction, are equal and opposite. 

If the portion of the wire which is doubled back be not left 
straight, but bent into any sinuosities, provided these be small 
compared with the distance between the wire and the frame, 
still no motion of the frame occurs when a current is set up in 
the wire. This is Ampére’s second case of equilibrium. It 
shows that the action of the elements of the curved conductor 
is the same as that of their projections on the straight conduc- 
tor. 

To obtain the ¢hird case of equilibrium, a wire, bent in the - 
arc of a circle, is arranged so that it may turn freely about a 
vertical axis passing through the centre of the circle of which 
the wire forms an arc, and normal to the plane of that circle. 
The wire is then free to move only in the circumference of that 
circle, or in the direction of its own length. Two vessels filled 
with mercury, so that the mercury stands above the level of 
their sides, are brought under the wire arc, and raised until 
conducting contact is made between the wire and the mercury 
in both vessels. A current is then passed through the mova- 
ble wire through the mercury. Then if any closed circuit 
whatever, or any magnet, be brought near the wire, it is found 


221] WaAGVEhLIO RELATIONS OF THE CURRENT. +13 


that the wire remains stationary. The deduction from this 
observation is that no closed circuit tends to displace an ele- 
ment of current in the direction of its length. 

In the fourth experiment three circuits are used, which we 
may call respectively 4, B,and C. They are alike in form, and 
the dimensions of & are mean proportionals to the correspond- 
ing dimensions of d and C. Lis suspended so as to be free 
to move, and 4 and C are placed on opposite sides of B, so 
that the ratio of their distances from # is the same as the 
ratio of the dimensions of A to those of &. If then the same 
current be sent through A and C, and any current whatever 
through JB, it is found that B does not move. The opposing 
forces due to the actions of A and C upon # are in equilib- 
rium. From this fourth case of equilibrium is deduced the 
law that the force between two current elements is inversely 
~as the square of the distance between them. 

Ampére made the assumption that the action between two 
current elements is in the line joining them. From the four 
-cases of equilibrium he then deduced an expression for the 
attraction between two current elements. It is 


£ oaety cos € — 3 cos 9 cos e’), (94) 


In this formula ds and ds’ represent the elements of the two 
circuits, 2 and z the strength of current in those circuits meas- 
ured in electromagnetic units, 7 the distance between the cur- 
rent elements, € the angles made by the two elements with one 
another, 6 and @ the angles made by ds and ds’ with 7 or r 
produced, the direction of the two elements being taken in the 
sense of their respective currents. 

A remarkable result of this equation is that two current 
elements of the same circuit in the same straight line repel 


By hehe me ELEMENTARY PHM SICS: [222 


one another. The angle € becomes = 0, and? =a amme 
Phe zt’ as ds’ 
therefore the force given by the equation is — Sere 


Since this is negative it expresses a repulsion. 

222. Solenoids and Electromagnets. — Ampére also 
showed that the action between two small plane circuits is the 
same as that between two small magnetic shells, and that a cir- 
cuit, or system of circuits, may be constructed which is the 
complete equivalent of any magnet. A long bar magnet may 
be looked on as made up of a great number of equal and simi- 
lar magnetic shells arranged perpendicular to the axis of the 
magnet, with their similar faces all in one direction. In order 
to produce the equivalent of this arrangement with the circuit, 
a long insulated wire is wound into a close spiral, straight and 
of uniform cross-section. The end of the wire is passed back 
through the spiral. When the current passes, the action of 
each turn of the spiral may be resolved into two parts, that 
due to the projection of the spiral on the plane normal to the 
axis, and that due to its projection on the axis. This latter 
component, for every turn, is neutralized by the current in the 
returning wire, and the action of the spiral is reduced to that 
of a number of similar plane circuits perpendicular to its axis. 
Such an arrangement is called a solenotd. The poles of a sole- 
noid of very small cross-section are situated at its ends, and 
it is equivalent to a bar magnet uniformly magnetized. 

If a bar of soft iron be introduced into the magnetic field 
within a solenoid it will become magnetized by induction. 
This combination is .called an electromagnet. Since the 
strength of the magnetic field varies with the strength of 
the current in the solenoid, and with the number of layers of 
wire wrapped around the iron core, the magnetization of any 
bar of iron whatever may be raised to its maximum by in- 
creasing the current or the number of turns of wire. 


224] WA sco ne LaA tl LONS OF THE CURRENT. 315 


223. Ampére’s Theory of Magnetism. — Ampére based 
upon these facts a famous theory of magnetism which bears 
his name. He assumed that around every molecule of iron 
there circulates an electrical current, and that to such molecular 
currents are due all magnetic phenomena. He made no hy- 
pothesis with regard to the origin or the permanency of these 
currents. The theory agrees with Weber’s hypothesis that 
maenetization consists in an arrangement of magnetic mole- 
cules. If we further adopt Thomson’s explanation of the dia- 
magnetic phenomena (§ 184), we may extend Ampére’s theory 
to all matter, and assume that an electrical current circulates 
about every molecule. In order to account for the different 
magnetic susceptibilities of different bodies, it must also be 
assumed that these molecular currents are of different intensi- 
ties in different kinds of matter. 

Ampeére’s theory, however, admits another explanation of 
diamagnetism, which was given by Weber. He assumes that 
all diamagnetic molecules are capable of carrying molecular 
currents, but that those currents, under ordinary conditions, 
do not exist in them. When, however, a diamagnetic body is 
moved up to a magnet, an induced current due to the motion 
(§ 226) is set up in each molecule, and in such a direction that 
the mojecules become elementary magnets, with their poles so 
directed towards the magnet in the field that there is repulsion 
between them. If this theory be true, it ought to be possible, 
as suggested by Maxwell, to lessen the intensity of magnetiza- 
tion of a body magnetized by induction, by increasing the 
strength of the field beyond a certain limit. 

224. The Hall Effect.—Hitherto it has been assumed that 
when currents interact, it is their conductors alone which are 
affected, and that the currents in the conductors are not in 
any way altered. Hall has, however, discovered a fact which 
seems to show that currents may be displaced in their conduc- 
tors. If the two poles of a voltaic battery be joined to two op- 


316 ELEMENTARY PHYSICS. [225 


posite arms of across of gold foil mounted on a glass plate, and 
if a galvanometer be joined to the other two arms. at such — 
points that no current flows through it, then if a magnet pole 
be brought opposite the face of the cross a permanent current 
will be indicated by the galvanometer. The same effect ap- 
pears in the case of other metals. The direction of the per- 
manent current and its amount differ under the same circum- 
stances for different metals. The coefficient which represents 
the amount of the Hall effect in any metal is called the vofa- 
tional coefficient of that metal. 

Since the rotational coefficients of such metals as have 
been tested agree in sign and in relative magnitude with their 
thermo-electric powers (§ 235), it is argued by Bidwell, Etting- 
hausen, and others that the Hall effect is due to thermo-electric 
action. 

225. Measurement of Current.—Instruments which are 
used to detect the presence of a current, or to measure its 
strength by means of the deflection of a magnetic needle, are 
commonly called galvanometers. 

The simplest form of the galvanometer is the old instru- 
ment called the Schwezgger's multiplier. It consists of a flat 
spool upon which an insulated wire is wound a number of 
times. The plane of the coils is vertical, and’ usually also co- 
incides with the plane of the magnetic meridian. A magnetic 
needle is suspended in the interior of the spool. When a cur- 
rent is passed through the wire, the needle is deflected from 
the magnetic meridian. Usually, in order to make the indica- 
tions of the apparatus more sensitive, a combination of two 
needles is used. They are joined rigidly together, so that 
when suspended the lower one hangs in the interior of the 
spool, and the other in the same plane directly above the 
spool. These needles are magnetized so that the positive end 
ot one is above the negative end of the other. If they are of 
nearly equal strength, such a combination will have very little 


225] WIAGNE TIC REEATIONS OF THE. CORRENT, 317 


directive tendency in the earth’s magnetic field. It is there- 
fore called an astatic system. When a current passes in the 
wire, however, the lines of force due to the current form closed 
curves passing through the coil, and both needles tend to turn 
in the same direction. Since the earth’s field offers almost no 
resistance to this tendency, an astatic system will indicate the 
presence of very feeble currents. The apparatus here described 
is no longer used to measure currents, but only to detect their 
presence and direction. 

The sxe galvanometer consists of a circular coil of insulated. 
wire, set in the vertical plane, in the centre of which is a sup- 
port for a magnetic needle. The needle can turn in the hori- 
zontal plane. When a current is sent through the coil, the 
magnet is deflected. The coil is then turned about the ver- 
tical axis, until the magnet lies in the plane of the coils. When 
this is the case, the equilibrium of the needle is due to the 
equality of the couples set up by the cur- 
rent in the coils and by the horizontal com- 
ponent of the earth’s magnetism. The couple 
due to the horizontal component (Fig. 68) is 
Hy sin @, where H represents the horizon- 
tal component, zz/ the magnetic moment of 
the magnet, and ¢@ the angle made by the 
plane of the coils with the magnetic meridian. 
The couple due to the current is, by Biot’s 
law, proportional to the current. It may then 
be set equal to kmz/, where & is a constant factor depending 
upon the dimensions of the galvanometer. Since these two 
couples are equal, we have the equation: 


Ss 


Sta eae 
¢= > sin p. (95) 


With the same galvanometer, then, different currents are pro- 


318 ELEMENTARY PHYSICS. : [225 


portional to the sines of the angles made with the magnetic 
meridian by the plane of the coils when the needle lies in that 
plane. If z be greater than — the equilibrium supposed in this 
explanation cannot occur. 

The tangent galvanometer is that form of galvanometer 
which is commonly used to measure current in electromagnetic 
units. It can best be discussed by considering first the action 
of a single circular current of strength z upon a magnet pole 
situated at any point on the normal to the plane of the circle 
drawn from its centre. 

The force due to any current element s (Fig. 69) upon 
mus 
ar 
This force tends to move the pole mw at right angles to the 
plane containing s and the line joining s and m. If we repre- 
sent by @ the angle between the line 
joining sand m and the # axis, the com- 


the magnet pole m, at the distance 4 is, by Biot’s law, —; 


Ss 


Mts 
ponents of this force become Te sin 


21S 

i cos 8 normal to 
the x axis. ‘The equal elements die 
metrically opposite s, also gives rise to 


m along the # axis, and - 


nits, 
Se sin @ along the + 


axis, which is added to the similar component due tto s, 
mts’ p eee 
and “7a COS 6 normal to the x axis, which is opposed to and 


Fic. 60. two components, 


annuls the similar component due to s. Every other similar 
pair of elements will give rise to two similar components along 
the x axis, and will annul one another’s action normal to the 
+ axis. The total force on # will then be a force along the 
+ axis equal to the sum of all the components along that axis. 


225] WinGNiniiG RELATIONS OF THE CURREN T. 319 


mis’ . 2rmir . : 
or ears sin 6, This equals jr sin 6, where 7 is the radius 
‘ ; e ; ‘ : 
of the circle. Since 7 = sin 6, this force may be written 
2mmir 2amir* 


amet heer 


If the circular coil considered be set vertical in the plane 
of the magnetic meridian, and a short magnetic needle be 
mounted at the point 7, so as to turn in the horizontal plane, 
the needle will be deflected from the meridian, and will rest in 
equilibrium between the force due to the current and that due 
to the earth’s magnetism. If the needle be so short that the dis- 
tance of its poles from the x axis may be neglected, the formula 
just obtained will give the force upon its poles. Let / repre- 
sent the half length of the needle (Fig. 70), @ its angle of 
deviation from the magnetic meridian, and d@ the 
distance from its centre to the plane of the coil. : 
Then d—/sin ¢ and d+ /sin®@ represent the |” cola 
distances of the magnet’s poles from the plane of e 
Mmiemeoiies the forces acting on these poles are f--"~"\-~ 
then | 

2amir* 2amir : 


CLa@—isingy °° CF GE Tan OE! pom 


If another precisely similar coil be set at the same distance d 
from the point of suspension of the needle, on the opposite 
side of it, and if the current be sent through it in the same 
direction, two other forces equal to those just stated will act 
upon the needle, tending to turn it in the same direction. 
There will thus arise two couples with moments equal to 


4umirl cos d 4nxmir'l cos 


(7? + (d+ Zsin d) 


320 ELEMENTARY PHYSICS. [225 


both tending to turn the magnet in the same direction. The 
I 


(7 + (d + /sin ¢))! 


factors are equal to 


(7? + a)" = 3° + a") *(2dlsin 6 +1 sin’ p) 
+ 15(77 + qd? Pee 17S (oe 


if we neglect all terms containing higher powers of @ than the 
second. In this expression the upper or the lower signs must 
be used throughout. When we add the two moments of couple, 
we obtain for the total moment of couple acting on the needle 
the expression, after reduction, 


raat 
LLU SA ase 3 eect — 4d 72 sin? #). 
TPR RN (ee 

This moment of couple is equal to that due to the horizon- 
tal intensity of the earth’s magnetism, or 2#H7/ sin ¢. Setting 
these expressions equal, we obtain for z, if we neglect powers 
of Z higher than the second, 


i Ebene ee, 
4m 


pay (t eon cnt @) (96) 


The best form of the tangent galvanometer is so constructed 


7 ‘ ‘ : 
that d= >. In this case the second term in the parenthesis 


< 
3 
3 


Jef 
disappears, and we have z = = ; a tang. The current is pro- 


~ 


portional to the tangent of the angle of deflection. If the 
aera coils contain a number of turns equal in each 


coil to = , the proportion of the breadth to the depth of the 


226] MAGNETIC RELATIONS OF THE CURRENT. 321 


coils may be so determined that the current is given by the 
equation 


; HR 
2= 16 ° ar tan —. (g7) 


In this equation A is the mean radius of the coil. All the 
quantities in this expression for z, except A, are either num- 
bers or lengths, and 7 can be measured in absolute units. The 
tangent galvanometer can therefore be used to measure current 
in absolute units. 

Weber's electro-dynamometer is an instrument with fixed 
coils like those of the tangent, galvanometer, but with a small 
suspended coil substituted for the magnet. The small coil is 
usually suspended by the two fine wires through which the 
current is introduced into it, and the moment of torsion of this 
so-called dzfilar suspension enters into the expression for the 
current strength. The same current is sent through the fixed 
and the movable coils, and a measurement of its strength can 
be obtained in absolute units, as with the tangent galvanome- 
ter. By a proper series of experiments, this measurement is 
made independent of the horizontal intensity of the earth’s 
magnetism. When the current is reversed in the instrument, 
the couple tending to turn the suspended coil does not change. 
If the effects of terrestrial magnetism can be avoided, the 
electro-dynamometer can therefore be used to measure rapidly 
alternating currents. 

226. Induced Currents.—It was shown in § 206 that the 
movement of a magnet in the neighborhood of a closed circuit 
will give rise, in general, to an electromotive force in the cir- 
cuit, and that the current due to this electromotive force will be 
in the direction opposite to that current which, by its action 
upon the magnet, would assist the actual motion of the mag- 
net. This current is called an zzduced current. From the 

21 


322 EBINGSMEN TARY VP SICS. 1226 


equivalence between a magnetic shell and an electrical cur- 
rent, it is plain that a similar induced current will be produced 
in a closed circuit by the movement near it of an electrical 
current or any part of one. Since the joining up or breaking 
«the circuit carrying a current is equivalent to bringing up that 
same current from an infinite distance, or removing it to an 
infinite distance, it is further evident that similar induced 
currents will be produced in a closed circuit when a circuit is 
made or broken in its presence. ; 

The demonstration of the production of induced currents 
in § 206 depends upon the assumption that the path of the 
magnet pole is such that work is done upon it by the current 


assumed to exist in the circuit. The potential of the magnet . 


pole relative to the current is changed. 

The change in potential from one point to another in the 
magnetic field due to a closed current is (Eq. 93) equal to 
7w@,— @-—+4an), and the work done on a magnet pole m, in 
moving it from one point to another, is 72z(@, — w-—+ 477). 
In the demonstration of § 206 we may substitute m(a,— @-+-472) 
for A, and, provided the change in potential be uniform, we 
mcr, — @ + 477) 

z 
tromotive force due to the movement of the magnet pole. 
If the change in potential be not uniform, we may conceive 
the time in which it occurs to be divided into indefinitely small 
intervals, during any one of which, ¢, it may be considered uni- 


obtain at once the expression — for the elec. 


form. Then the limit of the expression — ——+—-— : 


as ¢t becomes indefinitely small, is the electromotive force 


during that interval. 
The current strength due to this electromotive force is 


ma, — co + 472) 
Tula a 


226] MAGIVE dione LALION S: OF Tim CURRENT. 323 


If the induced current be steady, the total quantity of 
electricity flowing in the circuit is expressed by 


: 


m(co, — cot Ann) 
/ R ‘ 


The total quantity of electricity flowing in the circuit de- 
pends, therefore, only upon the initial and final positions of 
the magnet pole, and the number of times it passes through 
the circuit, and not upon its rate of motion. The electro- 
motive force due to the movement of the magnet, and conse- 
quently the current strength, depends, on the other hand, upon 
the rate at which the potential changes with respect to time. 

A more general statement, which will include all cases of 
the production of induced currents, may be derived by the use 
of the method of discussion given in § 219. The change in 
potential of a closed circuit, carrying a current in a magnetic 
field, may be measured by the change in the number of lines of 
force which pass through it in the positive direction. Any 
movement which changes the number of lines of force will set 
up in the circuit an electromotive force, and an induced current 
in a sense opposite to that current which would by its action 
assist the movement. As in the elementary case which has 
just been discussed, the total quantity of electricity passing 
in the circuit depends only upon the total change in the num- 
ber of lines of force passing through the circuit in the positive 
‘direction, but the electromotive force and current strength 
depend on the rate of change in the number of lines of force. 

It is often convenient, especially when considering the 
movement of part of a circuit in a magnetic field, to speak of 
the change in the number of lines of force enclosed by the 
circuit as the number of lines of force cut by the moving part 
of the circuit. The direction of the induced current in the 


324 ELEMENTARY PHYSICS; [226 


moving part of the circuit, if it be supposed to move normal 
to the lines of force, is related to the direction of motion and 
to the positive direction of the lines of force cut, in such a 
way that the three directions may be represented by the posi-, 
tive directions of the three co-ordinate axes of x, y, and 4, 
when the + axis represents the direction of motion, the y axis 
the lines of magnetic force, and the gz axis the direction of the 
induced current. The positive directions of the three axes are 
such that, if we rotate the positive 4 axis through a right angle 
about the z axis, clockwise as seen by one looking along the 
positive direction of the ¢ axis, it will coincide with the posi- 
tive y axis. 

The fact that induced currents are produced in a closed 
circuit by a variation in the number of lines of magnetic force 
included in it was first shown experimentally by Faraday in 
1831. He placed one wire coil, in circuit with a voltaic battery, 
inside another which was joined with a sensitive galvanometer. 
The first he called the primary, the second the secondary, cir- 
cuit. When the battery circuit. was made or broken, deflections 
of the galvanometer were observed. These were in such a 
direction as to indicate a current in the secondary coil, when 
the primary circuit was made, in the opposite direction to that 
in the primary, and when the primary circuit was broken, in 
the same direction as that in the primary. When the positive 
pole of a bar magnet was thrust into or withdrawn from the 
secondary coil, the galvanometer was deflected. The currents. 
indicated were related to the direction of motion of the posi- 
tive magnet pole, as the directions of rotation and propulsion 
in a left-handed screw. The direction of the induced currents 
in these experiments is easily seen 'to be in accordance with 
the law above stated, that the induced currents are always in 
the opposite direction to those currents which would, by 
their action, assist the motion. 

This law of induced currents in its general form was hxs = 


227| WAGNETIC RELATIONS OF THE CURRENT. 325 


announced by Lenz in 1834, soon after Faraday’s discovery of 
the production of induced currents. It is known as Levz’s law. 

The case in which an induced current in the secondary cir- 
cuit is set up by making the primary circuit is, as has been said, 
an extreme case of the movement of the primary circuit from 
an infinite distance into the presence of the secondary. The ex- 
periments of Faraday and others show that the total quantity of 
electricity induced when the primary circuit is made is exactly 
equal and opposite to that induced when the primary circuit is 
broken. They also show that the electromotive force induced 
in the secondary circuit is independent of the materials consti- 
tuting either circuit, and‘is proportional to the current strength 
in the primary circuit. These results are consistent with the 
formula already deduced for the induced current. 

227. Self-induction.—When a current is set up in any cir- 
cuit, the different parts of the circuit act on one another in the 
relation of primary and secondary circuits. Ina long straight 
wire, for example, the current which is set up through any 
small area in the cross-section of the wire tends to develop an op- 
posing electromotive force through every other areain the same 
cross-section. The true current will thus be temporarily weak- 
ened, and will require a certain time to attain its full strength. 
On the other hand, when the circuit is broken, the induced 
electromotive force is in the same direction as the electromo- 
tive force of the circuit. Since the time occupied by the change 
of the true current from its full value to zero, when the circuit 
is broken, is very small, the induced electromotive force is very 
great. The current formed at breaking is called the extra cur- 
vent, and gives rise to a spark at the point where the circuit is 
broken. The extra current may be heightened by anything 
which will increase the change in the number of lines of force, 
as by winding the wire in a coil and by inserting in the coil a 
piece of soft iron. This action ofa circuit onitself is called self 
anditction. 


326 ELEMENTARY PHYSICS. [228 


228. Electromagnetic Unit of Electromotive Force.—lf 
the circuit considered in § 226 move from a point where its po- 
tential relative to the magnet pole is ma, to one where it is 
mo, provided that the magnetic pole do not pass through the cir- 
cuit, and that the movement be so carried out that the induced 
current is constant, the electromotive force of the induced cur- 


m (G9, 


——F i ° . . 
rentis — sad If the movement take place in unit time, 


and if m(o,— @) also equal unity, the electromotive force in 
the circuit is defined to be uwuzt electromotive force. 

The expression m(@,— @) is equivalent to the change in 
the number of lines of force passing through the circuit in the 
positive direction. More generally, then, if a circuit or part of 
a circuit so move in a magnetic field that, in unit time, the 
number of lines of force passing through the circuit in the posi- 
tive direction increase or diminish by unity, at a uniform rate, 
the electromotive force induced is unit electromotive force. 

The simplest way in which these conditions can be presented 
is as follows: Suppose two parallel straight conductors at unit 
distance apart, joined at one end by a fixed cross-piece. Sup- 
pose the circuit to be completed by a straight cross-piece of unit 
length which can slide freely on the two long conductors. Sup- 
pose this system placed in a magnetic field of unit intensity, so 
that the lines of force are everywhere perpendicular to the 
plane of the conductors. Then, if we suppose the sliding piece 
to be moved with unit velocity perpendicular to itself along the 
parallel conductors, the electromotive force set up in the circuit 
will be the unit electromotive force. 

The unit of electromotive force thus defined is the electro- 
magnetic unit. In practice another unit is used, called the volt. 
It contains 10° C. G. S. electromagnetic units. 

To obtain the dimensions of electromotive force in the elec- 
tromagnetic system we need first the dimensions of number of 
linesof force. From the convention adopted by which lines of 


229] MAGNETIC RELATIONS OF THE CURRENT. 327 


force are used tomeasure thestreneth of a magnetic field we have 
n 

eq un ewmence (|= 173 14°77 Since, the electromo- 

tive force is measured by the rate of change of the number of 


lines of force we have [e] = ps cn" HIB Boek 


The definition of electromotive force is consistent, as it 
must be, with the equation ze = rate of work, or work divided 
by time. This equation is the same as that discussed in § 202, 
and holds whichever system of units is adopted. In the deter- 
mination of the unit of electromotive force the arrangement 
given above is, of course, impracticable. In those experiments 
which have been made, the induced electromotive force which 
was due to the rotation of a circular coil in a magnetic field 
was determined by calculation. 

229. Apparatus employing Induced Currents.—The pro- 
duction of induced currents by the relative movements of con- 
ductors and magnets is taken advantage of in the construction 
of pieces of apparatus which are of great importance not only 
for laboratory use but in the arts. . 

The telephonic receiver consists essentially of a bar magnet 
around one end of which is carried a coil of fine insulated wire. 
In front of this coil is placed a thin plate of soft iron. When 
the coils of two such instruments are joined in circuit by 
conducting wires, any disturbance of the iron diaphragm in 
front of one coil will change the magnetic field near it, and a 
current will be set up in the circuit. The strength of the mag- 
net in the other instrument will be altered by this current, and 
the diaphragm in front of it will move. When the diaphragm 
of the first instrument, or transmitter, is set in motion by sound- 
waves due to the voice, the induced currents, and the conse- 
quent movements of the diaphragm of the second instrument, or 
receiver, are such that the words spoken into the one can be 
recognized by a listener at the other. 


328 | ELEMENTARY PHYSICS. [229 


Other transmitters are generally used, in which the dia- 
phragm presses upon a small button of carbon. A current is 
passed from a battery through the diaphragm, the carbon but- 
ton, and the rest of the circuit, including the receiver. When 
the diaphragm moves, it presses upon the carbon button and 
alters the resistance of the circuit at the point of contact. This 
change in resistance gives rise to a change in the current, and 
the diaphragm of the receiver is moved. ‘The telephone serves 
in the laboratory as amost delicate means of detecting a change 
of current in a circuit. 

The various forms of magneto-electrical and dynamo-elec- 
trical machines are too numerous and too complicated for de- 
scription. In all of them an arrangement of conductors, usually 
called the armature, is moved in a pcwerful magnetic field, and 
a suitable arrangement is made by which the currents thus in- 
duced may be led off and utilized in an outside circuit. The 
magnetic field is sometimes established by permanent magnets, 


and the machine is called a magneto-machine. In most cases, — 


however, the circuit containing the armature also contains the 
coils of the electromagnets to which the magnetic field is due. 
When the armature rotates, a current starts in it, at first due to 
the residual magnetism of some part of the machine: this cur- 
rent passes through the field magnets and increases the strength 
of the magnetic field. This in turn reacts upon the armature, 
and the current rapidly increases until it attains a maximum 
due to the fact that the magnetic field does not increase pro- 
portionally to the current which produces it. Such a machine 
is called a dynamo-machine. 

The zuduction coil, or Ruhmkorff’s coil, consists of two cir- 
cuits wound on two concentric cylindrical spools. The inner 
or primary circuit is made up of a comparatively few layers of 
large wire, and the outer, or secondary, of a great number of 
turns of fine wire. Within the primary circuit is a bundle of 
iron wires, which, by its magnetic action, increases the electro- 


| 
: 


— ——— 


ee 


> 
a) Ve 


; 


230] MAGNETIC RELATIONS OF THE CURRENT. 329 


motive force of the induced current in the secondary coil. Some 
device is employed by which the primary circuit can be made 
or broken mechanically. The electromotive force of the induced 
current is proportional to the number of windings in the sec- 
ondary coil, and as this is very great the electromotive force of 
the induced current greatly exceeds that of the primary current. 
The electromotive force of the induced current set up when the 
primary circuit is broken is further heightened by a device pro- 
posed by Fizeau. To two points in the primary circuit, one on 
either side of the point where the circuit is broken, are joined the 
two surfaces of a condenser. When the circuit is broken, the 
extra current, if the condenser be not introduced, forms a long 
spark across the gap and so prolongs the fall of the primary cur- 
rent to zero. The electromotive force of the induced current is 
therefore not so great as it would be if the fall of the primary 
current could be made more rapid. When the condenser is in- 
troduced, the extra current is partly spent in charging the con- 
denser, the difference of potential between the two sides of the 
gap is not so great, the length of the spark and consequently 
the time taken by the primary current to become zero is 
lessened, and the electromotive force of the induced current is 
proportionally increased. 

230. Resistance.—As in the discussion of § 203, we may 
here define the ratio of the electromotive force to the current 
in any circuit as the resistance in that circuit. The electromag- 
netic unit of resistance is the resistance of that circuit in which 
unit electromotive force gives rise to unit current, when both 
these quantities are measured in electromagnetic units. In the 
example given in § 228, if we insert a galvanometer in that 
part of the circuit occupied by the fixed cross-piece, and 
assume that the resistance of every part of the circuit ex- 
cept the sliding piece is zero, the resistance of the sliding 
piece will be unity when, moving with unit velocity, it 
gives rise to unit current in the galvanometer. If it move with 


330 ELEMENTARY UP Y 3103. [230 


any other velocity v, and still produce unit current in the gal- 
vanometer, its resistance will be numerically equal to the veloc- 
ity v. For the electromotive force produced by a movement 
with that velocity is v, and the ratio of that electromotive force 
to unit current is v, which is the resistance by definition. 

A unit of resistance, intended to be the C.G.S. electromagnetic 
unit, was determined by a committee of the British Association 
by the following method: A circular coil of wire, in the centre 
of which was suspended a small magnetic needle, was mounted 
so as to rotate with constant velocity about a vertical diameter. 
From the dimensions and velocity of rotation of the coil and 
the intensity of the earth’s magnetic field, the induced electro- 
motive force in the coil was calculated. The current in the 
same coil was determined by the deflection of the small magnet. 
The ratio of these two quantities gave the resistance of the 
coil. 

In practice another unit of resistance is used, called the ohm. 
It would be the resistance of a sliding piece in the arrangement 
before described which would give rise to the C.G. S. unit of cur- 
rent if it were to move with a velocity of one billion centi- 
metres in a second. The ¢rue ohm thus contains 10° C.G.S. elec- 
tromagnetic units. The dimensions of resistance in the elec- 


e h 
tromagnetic system are [7] = eB —LT *. The dimensigas 


of resistance are therefore those of a velocity, as might be in- 
ferred from the measure of resistance in terms of velocity in 
the example given above. 

The standard of resistance, usually called the B.A. unit, de- 
termined by the committee of the British Association, has a 
resistance somewhat less than the true ohm as it is here defined. 
In practical work resistances are used which have been compared ° 
with this standard. The Electrical Congress of 1884 defined 
the “egal ohm to be “the resistance of a column of mercury of 
one square millimetre section and of 106 centimetres of length 


231| MAGNETIC RELATIONS OF’ THE CURRENT. Tae 


at the temperature of freezing.” The legal ohm contains 1.0112 
B. A. units. Boxes containing coils of wire of definite resistance, 
so arranged that by different combinations of them any desired 
resistance may be introduced into a circuit, are called reszstance 
boxes or rheostats. 

231. Kirchhoff’s Laws.—In circuits which are made up of 
several parts, forming what may be called a network of con- 
ductors, there exist relations among the electromotive forces, 
currents, and resistances in the different branches, which have 
been stated by Kirchhoff in a way which admits of easy appli- 
cation. | 

Several: conventions are made with regard to the positive and 
negative directions of currents. In considering the currents 
meeting at any point, those currents are taken as positive which 
come up to the point, and those as negative which move away 
from it. In travelling around any closed portion of the net- 
work, those currents are taken as positive which are in the di- 
rection of motion, and those as negative which are opposite to 
the direction of motion. Further, those electromotive forces are 
positive which tend to set up a positive current in their respec- 
tive branches. With those conventions Kzrchhoff’s laws may 
be stated as follows: 

1. The algebraic sum of all the currents meeting at any point 
of junction of two or more branches is equal to zero. This first 
law is evident, because, after the current has become steady, 
there is no accumulation of electricity at the junctions. 

2. The sum, taken around any number of branches forming 
a closed circuit, of the products of the currents in those branches 
into their respective resistances is equal to the sum of the elec- 
tromotive forces in those branches. This law can easily be 
seen to be only a modified statement of Ohm’s law, which was 
given in § 203. 

These laws may be best illustrated by their application in a 
form of apparatus known as Wheatstone's bridge. The circuit 


332 ' ELEMENTARY PAY SICS. [231 


of the Wheatstone’s bridge is made up of six branches. An 
end of any branch meets two, 
and only two, ends of other 
branches, as shown in Fig. 71. 
In the branch 6 is a voltaic cell 
with an electromotive force &. 
In the branch: 5 is a galvan- 
ometer which will indicate the 
presence of a current in that 
PiSar branch. In the other branches 
are conductors, the resistances of which may be called respec 
Lively My Wt Be iPas 
From Kirchhoff’s first law the sum of the currents meeting 
at the point C is z,+2,-+2,=0, and of those meeting at the 
point D isz,+72,+2,=0. By the second law, the sum of the 
products zy in the circuit ADC is 2,7, + 2,7, + 2,7, = 0, and in 
the circuit DBC is 2,7,-+-2,7,+2,7,=0, since there are no 
electromotive forces in those circuits. If we so arrange the 
resistances of the branches I, 2, 3, 4 that the galvanometer 
shows no deflection, then the current z, is zero, and these equa- 
tions give the relations, 2,=—42, %4,=— 2%, €7)== == 
2,7, = —2yr, From these four equations follows at once a 
relation between the resistances, expressed in the equation 


Pita vere. (98) 


If, therefore, we know the value of 7, and know the ratio of 7, 
to 7,, we may obtain the value of 7,. 

This method of comparing resistances by means of the 
Wheatstone’s bridge is of great importance in practice. By the 
use of a form of apparatus known as the Gratish Association 
bridge the method can be carried to a high degree of accuracy. 
In this form of the bridge, the portion marked ACB (Fig. 71) is 
a straight cylindrical wire, along which the end of the branch CD 


231] MAGNETIC RELATIONS OF THE CURRENT. 333 


is moved until a point C is found, such that the galvanometer 
shows no deflection. The two portions of the wire between C 
and A, and C and B, are then the two conductors of which the 
resistances are 7, and 7,, and these resistances are proportional 
to the lengths of those portions (§ 204). The ratio of 7, to 7, is 
therefore the ratio of the lengths of wire on either side of C, 
and only the resistance of 7, need be known in order to obtain 
mat ol 7; 

It is often convenient in determining the relations of current 
and resistance in a network of conductors to use Ohm’s law 
($203), directly, and consider the difference of potential between 
the two points on a conductor as equal to the product 77. 
When a part of a circuit is made up of several portions which 
all meet at two points A and #, the relation between the whole 
resistance and that of the separate parts may be obtained easily 
in this way. For convenience 
in illustration we will sup- 
pose the divided circuit (Fig. 
72) made up of only three 
portions, I, 2, 3, meeting at the 
points A and B, and that no electromotive force exists in those 
portions. Then the difference of potential between A and ZB is 
ee ee 27, 2,7, = 1,7,. We have also by Kirchhoff’s first 
law —7z,=2,+72,+2,. By the combination of these equations 
we obtain 


Fic. 72. 


Bee 21%) (+245). (99) 


The current in the divided circuit equals the difference of 
potential between A and & multiplied by the sum of the recip- 
rocals of the resistances of the separate portions. If we set this 


I : 7 HANTS ee 
sum equal to “sf and call 7 the resistance of the divided circuit, 


334 ELEMENTARY PHYSICS. [231 


we may say that the reciprocal of the resistance of a divided 
circuit is equal to the sum of the reciprocals of the resistances of 
the separate portions of the circuit. When there are only two 
portions into which the circuit is divided, one of them is usually 
called a shunt, and the circuit a shunt circuit. 

An arrangement devised by Clark, called the Clark's poten- 
tiometer, used to compare the electromotive forces of voltaic 
cells, depends for its action on the principles here discussed. 
It consists of a spiral of evenly drawn wire coiled about a rubber 
cylinder, with arrangements by which contact can be made with 
it at both ends and at any point along it. Let us call the cells 
to be compared cell 1 and cell 2, and let the electromotive force 
of cell 1 be the greater. To the two ends of the spiral are joined 
the terminals of a circuit which we will call A, containing a con- 
stant voltaic battery, of which the electromotive force is greater 
than that of either cell 1 or cell 2, and a set of. resistances which 
can be varied. To the same points are joined the terminals of 
a circuit which we will call 4, containing cell 1, and a sensitive 
galvanometer. The positive poles of the constant battery and 
of cell 1 are joined to the same end of the spiral. The resist- 
ance is then modified in circuit A until the galvanometer in 
circuit B shows no deflection. The difference of potential 
between the ends of the spiral is, therefore, equal and in the 
opposite direction to the electromotive force of cell 1. The 
positive pole of cell 2 is now joined to the end of the spiral to 
which the positive poles of the other circuits are joined, and 
with the free end of a circuit C, containing cell 2 and asensitive 
galvanometer, contact is made at different points on the spiral 
until the point is found at which, when contact is made, the 
galvanometer in C shows no deflection. The difference of poten. 
tial between that point and the end of the spiral joined to the 
positive poles is equal and opposite to the electromotive force 
of cell 2. The electromotive forces of the two cells are then 
proportional to the lengths of the wire between the points of 


231] MAGNETIC RELATIONS OF THE CURRENT. 335 


contact of their terminals; that is, the electromotive force of 
cell 1 is to that of cell 2 as the length of the wire spiral is to 
that portion of its length between the two terminals of cell 2. 
For, since the wire is uniform, its resistance is proportional to 
its length, and if we represent the potential of the common 
point of contact of the positive poles by V, the potentials of 
the points of contact of the two negative poles by V, and V,, 
the current in the spiral by z, and the resistances of the lengths 
of wire considered by 7, and 7,, we have 


The rules for joining up sets of voltaic cells in circuits so as 
to accomplish any desired purpose may be discussed by the 
same method. Let us suppose that there are x cells, each with 
an electromotive force e and an internal resistance 7, and that 
the external resistance of the circuit iss. If # be a factor of 
m, and if we join up the cells with the external resistance so as 
to form a divided circuit of parallel branches, each containing 


n debe 
pa cells, we shall have for the electromotive force in sucha 


ur 
2 


; ya BE : ; ; 
circuit —, and for the resistance of the circuit s + The 
m 


m1 
current in the circuit is therefore 7=————- 

mis t-ur 
may arise which are common in practice. The resistance s of 
the external circuit may be so great that, in comparison with 
m’s, ur may be neglected. In that case z isa maximum when 
m = 1, that is, when the cells are arranged tandem, or in series, 
with their unlike poles connected. On the other hand, if 72s 
be very small as compared with m7, it may be neglected, and z 
becomes a maximum when 7 = z, that is, when the cells are 


Two cases 


| 


336 ELEMENTARY PHYSICS. [233 


arranged abreast, or in multiple arc, with their like poles in con- 
tact. 

232. Ratio between the Electrostatic and Electromag- 
netic Units.—When the dimensions of any electrical quantity 
derived from its electrostatic definition are compared with its 
dimensions derived from its electromagnetic definition, the 
ratio between them is always of the dimensions of some power 
of a velocity. The ratio between the electrostatic and electro- 
magnetic unit of any electrical quantity is, therefore, of the 
dimensions of some power of a velocity. If, therefore, this 
ratio be obtained for any set of units, the number expressing it 
will also express some power of a velocity. This velocity is an 
absolute quantity or constant of nature. Whatever changes 
are made’in the units of length and time, the number express- 
ing this velocity in the new units will also express the ratio of 
the two sets of electrical units. 

This ratio, which is called v, can be measured in several 
ways. 

The first method, used by Weber and Kohlrausch, depends 
upon the comparison of a quantity of electricity measured in 
the two systems. From the dimensions of current in the elec- 
tromagnetic system we have the dimensions of quantity 
[7] =[¢7] = A*L?. The dimensions of quantity in the electro- 
static system are[Q] = M?L?7". The ratio of these dimen- 


Ary bees : 
sions 1s bse = L 7", or, the number of electrostatic units of 
qd 


quantity in one electromagnetic unit is the velocity w. 

In Weber and Kohlrausch’s method the charge of a Leyden 
jar was measured in electrostatic units by a determination of 
its capacity and the difference of potential between its coatings. 
The current produced by its discharge through a galvanometer 
was used to measure the same quantity in electromagnetic 
measure. 

Thomson determined v by a comparison of an electromotive 


232] MAGNETIC RELATIONS OF THE CURRENT. 527, 


force measured in the two systems. Hesent a current through 
a coil of very high known resistance, and measured it by an 
electro-cdynamometer. The electromagnetic difference of po- 
tential between the two ends of the resistance coil was then 
equal to the product of the current by the resistance. The 
electrostatic difference of potential between the same two points 
was measured by an absolute electrometer. From the dimen- 
sional formulas we have 


aA Gay ema 
fiona sit 


The number of electromagnetic units of electromotive force 
in one electrostatic unit is v. The ratio of the numbers express- 
ing the electromagnetic and the electrostatic measures of the 
electromotive force in Thomson’s experiment is therefore the 
quantity v. ‘This experiment was carried out by Maxwell ina 
different form, in which the electrostatic repulsion of two simi- 
larly charged disks was balanced by an electromagnetic attrac- 
tion between currents passing through flat coils on the back of 
the two disks. 

Other methods, depending on comparisons of currents, of 
resistances, and other electrical quantities, have been employed. 
The methods described are historically interesting as being the 
first ones used. The values of v obtained by them differed 
rather widely from one another. Recent determinations, how- 
ever, give more consistent results. It is found that v, considered 
as a velocity, is about 3-10°° centimetres in a second. This 
velocity agrees very closely with the velocity of light. 

The physical significance of this quantity v may be under- 
stood from an experiment of Rowland. The principle of the 
experiment is as follows. If we consider an indefinitely ex- 


tended plane surface on which the surface density of electrifica- 
22 


338 ELEMENTARY PHYSICS. 232] 


o 
tion is ©, measured in electrostatic units, or — measured in elec- 
| v 


tromagnetic units, since the ratio of the electrostatic to the 
electromagnetic unit of quantity is v; and conceive it to move 
in its own plane with a velocity +; the charge moving with it 
may be considered as the equivalent of a current in that sur- 
face, the strength of which, measured by the quantity of elec- 
tricity which crosses a line of unit length, perpendicular to the 


: ee Pg 6.7 
direction of movement, in unit time, is —- The force due to 
v 


such a current on a magnet may be calculated. Conversely, if 
the force on the magnet be observed, and the surface density 
o and the velocity x be also measured, the value of v may be 
calculated. The probability of such an action as the one here 
described was stated by Maxwell. 

The experiment by which Rowland verified Maxwell's view 
consisted in rotating a disk cut into numerous sectors, each of 
which was electrified, under an astatic magnetic needle. Dur- 
ing the rotation of the disk, a deflection of the needle was ob- 
served, in the same sense as that in which it would have moved 
if a current had been flowing about the disk in the direction of 
its rotation. From the measured values of the deflecting force, 
of the surface density of electrification on the disk, and the 
velocity of rotation, Rowland calculated a value of wv which lies 
between those given by Weber and Maxwell. 

It may be seen that, if the velocity + of the moving surface 
which we at first considered be equal to v, the equivalent cur- 
rent strength in the surface will be o. If we imagine another 
such surface near the one already considered, the repulsion be- 
tween them due to their opposite charges is 270° for every unit 
of surface (§ 198). It can be shown, by a method too extended 
to be given here, that the attraction between two currents in 
the same surfaces, of which the strengths in the surface are both 
og, is also expressed by 270” for every unit of surface. Hence 


MAGNETIC RELATIONS OF THE CURRENT. 339 


Mi Pri surfaces, so charged that the surface density of their elec- 
7 trification is o, can move with a velocity in their own planes 
equal to v, the repulsion of the charges will exactly counter- 


i ¥ balance the attraction of the currents due to their movement. 


Hed 


7 
oe 1 (" 
|, 


CHE PAE eval 


THERMO-ELECTRIC RELATIONS OF THE CURRENT, 


233. Thermo-electric Currents.—The heating or cooling 
of a junction of two dissimilar metals by the passage of a 
cu.rent, referred to in § 200 as the Peltier effect, is the reverse 
of a phenomenon discovered in 1822-23 by Seebeck. He 
found that, when the junction of two dissimilar metals was 
heated, a current was sent through any circuit of which they 
formed a part. It has since been shown that the same phe- 
nomenon appears if the junction of two liquids, or of a liquid 
anda metal, be heated. This fact, as has been already shown 
in § 206, follows as a result of the Peltier phenomenon. If 


we designate by P the heat developed at the junction by the. 


passage of unit current for unit time, we may substitute it for 


of USI, ; ; 
the expression rte the general equation of § 206, and obtain 


ie eae The counter electromotive force set up at the 
heated junction is the coefficient P, and is the measure of the 
true electromotive force of contact ($ 214). The contact elec- 
tromotive force of Volta does not agree in magnitude and not 
always in sign with this electromotive force. From this fact 
it is evident that the contact electromotive force of Volta is 
at least partially due to the air or other medium in which the 
bodies which are tested are placed. 

If the electromotive force & and the current J be reversed 


in the circuit, the junction is cooled and we obtain / = sa 
X 


—_ 


234] THERMO-ELECTRIC RELATIONS OF THE CURRENT. 341 


The electromotive force at the junction, therefore, tends to 
increase the electromotive force of the circuit. Since this is 
opposite to the electromotive force of the circuit in the case 
in which the junction is heated, the direction of the electro- 
motive force at the junction is the same as that found in the 
other case. If, then, there be no electromotive force £ in the 


ame a 
circuit, we have / = — R in case a unit of heat is communi- 
cated to the junction and absorbed by it in unit time, and: 
saith ies 
U igan 7p in case a similar quantity of heat is removed from the 


junction by cooling. | 

If two strips of dissimilar metals, for example antimony 
and bismuth, be placed side by side, and united at one end 
of the pair, being everywhere else insulated from one another, 
the combination is called a thermo-electric element. If several 
such elements be joined in series, 
so that their alternate junctions 
lie near together and in one plane, 
as indicated in Fig. 73, such an 
arrangement is called a ¢hermo- 
pile. When one face of the pile 
is heated, the electromotive force 
of the pile is the sum of the elec- Fic. 73. 
tromotive forces of the several elements. Such an instrument 
was used by Melloni, in connection with a delicate galvanom- 
eter, in his researches on radiant heat. 

When a thermo-electric element is constructed of any two 
metals, that metal is said to be thermo-elcctrically positive to the 
other from which the current flows across the heated junction. 

234. Thermo-electric Series.—It was found by the experi- 
ments of Seebeck himself, and those of others, that the metals 
may be arranged in a series such that any metal in it is thermo- 


342 ELEMENTARY PHYSICS. [235 


electrically positive to those which follow it, and thermo-elec- 
trically negative to those which precede it. | 

If a circuit be formed of any two metals in this series, and 
one of the junctions be kept at the temperature zero, while the 
other is heated to a fixed temperature, there will be set up an 
electromotive force which can be measured. If now the circuit 
be broken at either junction, and the gap filled by the intro- 
duction of any other metals of the series, then, provided that 
the junction which has not been disturbed be kept at the tem- 
perature which it previously had, and that the other junctions 
in the circuit be all raised to the temperature of the junction 
which was broken, there will be the same electromotive force 
in the circuit as existed before the introduction of the other 
metals of the series. It is manifest, then, that in a circuit made 
up of any metals whatever, at one temperature, no electromo- 
tive force can be set up by changing the temperature of the 
circuit as a whole. 

Thomson showed that it is not necessary for the production 
of thermal currents that the circuit should contain two metals: 
but that want of homogeneity arising from any strain of one 
part of an otherwise homogeneous circuit will also admit of the 
production of such currents. It has also been shown that when 
a portion of an iron wire is magnetized, and is heated near one 
of the poles produced, a thermal current will be set up. 

Cumming discovered in 1823 that, if the temperature of one 
junction of a circuit of two metals be gradually raised, the cur- 
rent produced will increase to a maximum, then decrease until 
it becomes zero, after which it is reversed and flows in the 
opposite direction. The experiments of Avenarius, Tait, and 
Le Roux show that, for almost all metals, the temperature of 
the hot junction at which the maximum current occurs is the 
mean between the temperatures of the two junctions at which 
the current is reversed. 

235. Thermo-electric Diagram.—The facts hitherto dis- 
covered in relation to thermo-electricity may be collected in a 


235| 7HERMO-ELECTRIC RELATIONS OF THE CURRENT. 343 


general formula or exhibited by means of a thermo-electric dia- 
gram. 

Let us consider a circuit of two metals, copper and lead, in 
which both junctions are at first at the same temperature. We 
may assume that there is an equal electromotive force of contact 
at both junctions acting from lead to copper. If one of the 
junctions be gradually heated, a current will be set up, passing 
from lead to copper across the hot junction. The heating has 
disturbed the equilibrium of electromotive forces, and has in- 
creased the electromotive force across the hot junction from 
lead to copper. The rate at which this electromotive force 
changes with change in the temperature is called the ¢hermo- 
electric power of the two metals. ‘That is, if A represent the 
electromotive force, ¢ the temperature, and @ the thermo- 


' —£E 
electric power, we have ——-—— 


1 0 


z, are indefinitely near one another. Hence if we lay off on 
the axis of abscissas (Fig. 74) an infinitesimal length 7, —- ¢,, and 
erect as ordinate the corresponding thermo-electric power 6, 
the area of the rectangle formed by the two lines will represent 
the electromotive force £, — £,, due to the change in tempera- 
ture. If, beginning at the point z,, we lay off the similar infini- 
tesimal length ¢, — z,, and erect as ordinate the thermo-electric 
power 6,, we shall obtain another rectangle representing the 
electromotive force £,— &,. So for any temperature changes 
the total area of the figure 3, 
bounded by the axis of tem- 


0 in the limit where # and 


_ 
—_— — 
— = 
—™ == oe 
— = an 
— aa 
_— 


y 
peratures, by the ordinates ! Pi 
representing the thermo-elec- a! 
tric powers at the temper- 
atures ¢, and ¢,, and by the 
curve AA’ passing through 44% ty 


the summits of the rectangles Fic. 74. 
so obtained, will represent the electromotive force due to the 
heating of the junction from Z, to ¢,. 


344 ELEMENTARY PHYSICS. [235 


It was found by Tait and Le Roux that the thermo-electric 
power, referred to lead as a standard, of all metals but iron and 
nickel, is proportional to the rise in temperature. The curve 
AA’ is therefore for those metals a straight line. For iron and 
nickel the curve is not straight. 

For another metal in comparison with lead, the line BA’, cor- 
responding to the line AA’ for copper, may have a different 
direction. From what has been said about the possibility of 
arranging the metals in a thermo-electric series, it is evident 
that the thermo-electric power between copper and the other 
metal is the difference of their thermo-electric powers referred 
to lead, and that the electromotive force at the junction of the 
two metals, due to a rise of temperature from 7¢, to ¢,, is repre- 
sented by the area of the figure contained by the two terminal 
ordinates and the two lines AA’ and BA’. The thermo-elec- 
tric power is reckoned positive when the current sets from lead 
to copper across the hot junction. In the diagram the ther- 
mo-electric power A’S’ is positive, and the electromotive force 
indicated by the area is from copper to the other metal across 
the hot junction. At the point where the lines 44’ and BB’ 
intersect, the thermo-electric power for the two metals vanishes. 
The temperature at which this occurs is called the neutral 
temperature and is designated by ¢,. When the temperature 
Z, lies on the other side of the neutral temperature from /,, the 
thermo-electric power becomes 
negative, and the electromotive 
force due to the rise in tempera- 
ature from ¢, to ¢, is negative. In 
Fig. 75 it is at once seen thas 
A’'B’ is negative for ¢,, and that 
the area VA’S’ is also negative. 
The electromotive force due toa 
rise of temperature from ¢/, increases until the temperature of 
the hot junction is ¢,, when it is a maximum, and then de. 


Fic, 75. 


m35|7HERMO-ELECTRIC RELATIONS OF THE CURRENT, 345 


creases. When the area WVA’S’ becomes equal to the area 
AWB, the total electromotive force is zero; when VA’#’ is 
greater than AVS, the electromotive force becomes negative, 
and the current is reversed. In case AA’ and BH’ are straight 
lines it is plain that the temperature z,, at which this reversal 
occurs, will be such that the neutral temperature ¢,.1s a mean 
between /, and 7,. 

The same facts can be represented by a general formula. 
Thomson first pointed out that the fact of thermo-electrical in- 
version necessitates the view that the thermo-electric power at 
a junction is a function, of the temperature of that junction. 
Avenarius embodied this idea in a formula, which his own re- 
searches, and those of Tait, show to be closely in agreement 
with experiment. Let us call the hot junction 1 and the cool 
junction 2, and set the electromotive force at each junction as 
a quadratic function of the absolute temperatures. We have 
fo=A+b4,+ ct? and £,=A-+ 6t,+ ct, where A, 0, and 
¢are constants. The difference £, — £,, or the electromotive | 
force in the circuit, is 


&, — E, = b(¢, — t,) + ¢ (4? — 7,’) 
as (4, ie zt) (0 aT c(Z, ae t,)) 
This equation may be put in the form used by Tait, if we 


a 
write 6= at, and¢ = — es We then have 


E, — £,= a(t, — ¢,) (tn — 44, +24) (100) 


The electromotive force in the circuit can become zero 
when either of these terms equals zero. It has been already 
stated that when ¢, =—7,, or when both junctions are at the 
same temperature, there is no electromotive force in the circuit. 


340 ELEMENTARY PHYSICS. [235 


When 4(¢, + ¢,) equals z,, or when the mean of the tempera- 
tures of the hot and cold junctions equals a certain temperature, 
constant for each pair of metals, there will be also no electro. 
motive force in the circuit. This temperature ¢, is that which 
has already been called the neutral temperature. The formula 
also assigns the value to that temperature ¢, at which, for fixed 
values of ¢, and ¢,, the electromotive force in the circuit is a 
maximum. If we represent the difference between ¢, and z, by 
x,thenz,=—7, +. Using this value in the formula, we ob- 


tain 2, — 4, ~ (¢, —Z,) —2x°). This is manifestly a max- 


imum when+=0. Theelectromotive force ina circuit is then, 
according to the formula, a maximum when the temperature of 
one junction is the neutral temperature. : 

The formula also shows that the thermo-electric power is 


a 
zero when ¢#, =7,. We may set &, = A+ at,t, — i t,’. Now 


if ¢, take any small increment 47,, &, has a corresponding in- 
crement Z&,. Hence we have 


E,+4E,=A-+ att, —* t+ at, 4t, — at, At, 


if we neglect the term containing 47,7, From this equation 


we obtain aR = at, — at,, which in the limit, as 47, becomes 


indefinitely small, is the thermo-electric power at the tempera- 
ture 7. Itis positive for values of ¢, below ¢,; 1s Zero for, 
= ?,, and negative for higher values of ¢,. That is, if we as- 
sume ¢, = Z, lower than ¢,, and then gradually raise the tem- 
perature ¢,, the thermo-electric power at the heated junction is 
at first positive, but continually decreases in numerical value, 
until at 7, = ¢, it becomes zero. At that temperature, then, 
the metals are thermo-electrically neutral to one another, and a 


236] 7HERMO-ELECTRIC RELATIONS OF THE CURRENT. 347 


small change in the temperature does not change the electro- 
motive force at the junction. 

236, The Thomson Effect.—Thomson has shown that, in 
certain metals, there must be a reversible thermal effect when 
the current passes between two unequally heated parts of the 
same metal. Let us suppose a circuit of copper and iron, of 
which one junction is at the neutral temperature, and the other 
below the neutral temperature. The current then sets from 
copper to iron across the hot junction. In the hot junction 
there is no thermal effect produced, because the metals are at 
the neutral temperature. Across the cold junction the current 
is flowing from iron to copper, and hence is evolving heat. The 
current in the circuit can be made to do work, and since no 
other energy is imparted to the circuit this work must be done 
at the expense of the heat in the circuit. Since heat is not 
absorbed at either junction, it must be absorbed in the unequally 
heated parts of the circuit between the junctions. 

To show this, Thomson used a conductor the ends of which 
were kept at constant temperatures in two coolers, while the 
central portion was heated. When acurrent was passed through 
this conductor, thermometers, placed in contact with exposed 
portions of the conductor between the heater and the coolers, 
indicated a rise of temperature different according as the cur- 
rent was passing from hot to cold or from cold to hot. The 
heat seems therefore to be carried along by the current, and the 
process has accordingly been called the electrzcal convection of 
feat. In copper the heat moves with the current, in iron 
against it. In another form of statement, it may, be said that, 
in unequally heated copper, a current from hot to cold heats 
the metal, and from cold to hot cools it, while in iron the 
reverse thermal effects occur. The experiments of Le Roux 
show that the process of electrical convection of heat cannot be 
detected in lead. For this reason, lead is used as the standard 
metal in constructing the thermo-electric diagram. 


CHAPT ERNIE 
LUMINOUS EFECTS OF THE CURRENT. 


237. The Electric Arc.—If the terminals of an electric 
circuit in which is an electromotive force of forty or more volts 
be formed of carbon rods, a brilliant and permanent luminous 
arc will appear between the ends of the rods if they be touched 
together and then withdrawn a short distance from each 
other. The temperature of the arc is so high that the most 
refractory substances melt or, are dissipated when placed in it. 
The carbon forming the positive terminal is hotter than the 
other. Both the carbons are gradually oxidized, the loss of the 
positive terminal being about twice as great as that of the nega- 
tive. The arc is, however, not due to combustion, since it can 
be formed in a vacuum. 

The current passing in the arc is, in ordinary cases, not 
sreater than ten amperes, while the measurements of the resist- 
ance of the arc show that it is altogether too small to account 
for this current when the original electromotive force is taken 
into account. This fact has been explained by Edlund and 
others on the hypothesis that there is a counter electromotive 
force set up in the arc, which diminishes the effective electro- 
motive force.of the circuit. The measurements of Lang show 
that this counter electromotive force in an arc formed between 
carbon points is about thirty-six volts, and in one formed be- 
tween metal points about twenty-three volts. 

238. The Spark, Brush, and Glow Discharges.—When 
a conductor is charged to a high potential and brought near an- 
other conductor which is joined to ground, a spark or a series 


238] LUMINGW STEEP FECT S OF (THE) CURRENT. 349 


of sparks will pass from one to the other. This phenomenon 
and others associated with it are most readily studied by the 
use of an electrical machine or an induction coil, between the 
electrodes of which a great difference of potential can be easily 
produced. If the spark be examined with the spectroscope, its 
spectrum is found to be characterized by lines which are due to 
the metals composing the electrodes, and to the medium between 
them. 

The passage of the spark through air or any dielectric is 
attended with a sharp report, and if the dielectric be solid, it 
is perforated or ruptured. If the electrodes be separated by a 
considerable distance, the path of the spark is usually a zigzag 
one. It is probable that this is due to irregularities in the 
dielectric, due to the presence of dust particles. 

With proper adjustment of the electrodes, the discharge 
may sometimes be made to take the form of a long drush spring- 
ing from the positive electrode, with a single trunk which 
branches and becomes invisible before reaching the negative 
electrode. Accompanying this is usually a number of small and 
irregular brushes starting from the negative electrode. 

Another form of discharge consists of a pale luminous g/ow 
covering part of the surface of one or both electrodes. If a 
small conducting body be interposed between the electrodes 
when the glow is established, a portion of the glow will be cut 
off, marking out a region on the electrode which is the projec- 
tion of the intervening conductor by the lines of electrical 
force. This phenomenon is called the electrical shadow. 

The difference of potential required to set up a spark be. 
tween two slightly convex metallic surfaces, separated by a 
stratum of air 0.125 centimetres thick, has been shown by 
Thomson to be about 5500 volts. The difference of potential 
which produces the sparks between the electrodes of an elec- 
trical machine, which are sometimes fifty or sixty centimetres 
long, must therefore be very great. The quantity of electricity 


350 ELEMENTARY PHYSICS. [239 


which passes during the discharge is, however, exceedingly 
small, on account of the great resistance of the medium through 
which the discharge takes place. 

Faraday showed that many of the phenomena of the dis- 
charge depend to some extent upon the medium in which it 
occurs. The differences in color and in the facility with which 
various forms of the discharge were set up in the gases upon 
which he experimented were especially noticeable. 

It was proved by Franklin that the lightning flash is an 
electrical discharge between a cloud‘and the earth or another 
cloud at a different electrical potential. The differences of 
potential to which such discharges are due must be enormous, 
and the heat developed by the discharge shows that the quantity 
of electricity which passes in it is not inconsiderable. 

Slowly moving fire-balls are sometimes seen, which last for a 
considerable time and disappear with a loud report and with 
all the attendant phenomena of a lightning discharge. It is 
not improbable that they are glow discharges which appear 
just before the difference of potential between the cloud and 
the earth becomes sufficiently great to give rise to a lightning 
flash. 

239. The Electrical Discharge in Rarefied Gases.—lIf the 
air between the electrodes of an electrical machine be heated, 
it is found that the discharge takes place with greater facility 
and that the spark which can be obtained is longer than before. 
Similar phenomena appear if the air about the electrodes be 
rarefied by means of an air-pump. -After the rarefaction has 
reached a certain point the discharge ‘ceases to pass as a spark 
and becomes continuous. The arrangement in which this dis- 
charge is studied consists of a glass tube into which are sealed two 
platinum or, preferatly, aluminium wires to serve as electrodes, 
and from which the air is removed to any required degree of 
exhaustion by an air-pump. Such an arrangement is usually 
called a vacuum-tube. 


239] LUMINOUS EFFECTS OF THE CURRENT. S01 


As the exhaustion proceeds there appears about the negative 
electrode in the tube a bright glow, separated from the 
electrode by a small non-luminous region. The body of the 
tube is filled with a faint rosy light, which in many cases breaks 
up into a succession of bright and dark layers transverse to the 
direction of the discharge. The discharge in this case is called 
the stratified discharge. A vacuum-tube in which the exhaus- 
tion is such that the phenomena are those here described is 
often called a Gezssler tube. As the exhaustion is raised still 
higher, the rosy light in the tube fades out, the non-luminous 
space around the negative electrode becomes very much greater, 
and the phenomena in the tube become exceedingly interesting. 
They were discovered and have been carefully studied by 
Crookes, and the vacuum-tubes in which they appear are hence 
called Crookes tubes. They may be most conveniently de- 
scribed by assuming that there is a special discharge from the 
negative electrode, which we will usually call the discharge. 
‘This view receives some support from the fact that the relations 
of current and resistance in the tube are such as to indicate a 
counter electromotive force at the negative electrode. 

The region occupied by the discharge from the negative 
electrode may be recognized by a faint blue light, which was 
not visible in the former condition of the tube. At every point 
on the wall of the tube to which this discharge extends occurs 
a brilliant -hosphorescent glow, the color of which depends on 
the nature of the glass. The discharge seems to be indepen- 
dent of the position of the positive electrode, and to take place 
in nearly straight lines, which start normally from the negative 
electrode. If two negative electrodes be fixed in the tube, the 
discharge from one seems to be deflected by the other, and two 
discharges which meet at right angles seem to deflect one 
another. 

If the discharge from a flat electrode be made to fall upon a 


352 ELEMENTARY PHYSICS. [239 


body which can be moved, such as a glass film, or the vane of 
a light wheel, mechanical motions will be set up. 

If the negative electrode be made in the form of a spherical 
cup, and a strip of platinum foil be placed at its centre, the foil 
will become heated to redness when the discharge is set up. 

Two discharges in the same direction repel one another as if 
they were similarly electrified, and a magnet, brought near the 
outside of the tube, will deflect a discharge as if it were an 
electrical current. 

The explanation of these phenomena is probably that given 
by Crookes, and adopted by Spottiswoode and Moulton. Itis 
assumed that they are due to the presence of the molecules of 
gas left in the tube after the exhaustion has been brought to 
an end. The mean free path of the molecules in the tube is 
much greater than that at ordinary densities, and they can 
accordingly move through long distances in the tube before 
their original motion is checked by collision with other mole- 
cules. It is assumed that the molecules of gas in the tube are 
attracted by the negative electrode, are charged negatively by 
it, and are then repelled. The phenomena which have been 
described are then due to their collision with other bodies or 
with the wall of the tube, or to their mutual electrical repul- 
sions and to the action between a moving quantity of electricity 
and a magnet. 

The experiments of Spottiswoode and Moulton, who showed 
that the same phenomena appeared at lower exhaustions, if the 
intensity of the discharge were increased, are in favor of this 
explanation. So is also the fact that the Crookes phenomena 
appear with a maximum intensity at a certain period during the 
exhaustion of the tube, while if the exhaustion be carried as 
far as possible, by the help of chemical means, they cease 
altogether and no current passes in the tube. The connection 
of these phenomena with the action of the radiome*cr (§156) 
is also at once apparent. 


SOUND. 


CHAPEERS I, 
ORIGIN AND TRANSMISSION OF SOUND. 


240. Definitions.—Acoustics has for its object the study of 
those phenomena which may be perceived by the ear. The 
sensations produced through the ear, and the causes that give 
rise to them, are called sounds. 

241. Origin of Sound.—Sound is produced by vibratory 
movements in elastic bodies. The vibratory motion of bodies © 
when producing sound is often evident to the eye. In some 
cases the sound seems to result from a continuous movement, 
but even in these cases the vibratory motion can be shown by 
means of an apparatus known as a manometric capsule, devised 
by Konig. It consists of a block A, Fig. 76, 
in which is a cavity covered by a membrane 
6. By means of a tube c illuminating gas is 
led into the cavity, and, passing out through 
the tube d, burns ina jet ate. It is evident | 
that, if the membrane 6 be made to move Fic. 76. 
suddenly inward or outward, it will compress or rarefy the gas 
in the capsule, and so cause the flow at the orifice and the 
height of the flame to increase or diminish. Any sound of 
sufficient intensity in the vicinity of the capsule causes an al- 
ternate lengthening and shortening of the flame, which, how- 
ever, occur too frequently to be directly observed. By mov- 

23 


354 ELEMENTARY PHYSICS. [242 


ing the eyes while keeping the flame in view, or by observing 
the image of the flame in a mirror which is turned from side 
to side, while the flame is quiescent, it appears drawn out into 
a broad band of light, but when it is agitated by a sound near 
.it, it appears serrate on its upper edge or even as a series of 
separate flames. ‘This lengthening and shortening of the flame 
is evidence of a to-and-fro movement of the membrane, and 
hence of the sounding body that gave rise to the movement. 
If a hole be made in the side of an organ-pipe and the capsule 
made to cover it, the vibrations of the air-column within the 
pipe may be shown. By suitable devices the vibratory motion 
of all sounding bodies may be demonstrated. 

242. Propagation of Sound.—The vibratory motion of a 
sounding body is ordinarily transmitted to the ear through the 
air. This is proved by placing a sounding body under the re- 
ceiver of an air-pump and exhausting the air. The sound be- 
comes fainter and fainter as the exhaustion proceeds, and 
finally becomes inaudible if the vacuum is good. Sound may, 
however, be transmitted by any elastic body. 

In order to study the character of the motion by which 
sound is propagated, let us suppose AB (Fig. 77) to represent 


Fre. 77. 


a cylinder of some elastic substance, and suppose the layer of 
particles a to suffer a small displacement to the right. The 
effect of this displacement is not immediately to move forward 
the succeeding layers, but a approaches 6, producing a conden- 
sation, and developing a force that soon moves 6 forward; this 
in turn moves forward the next layer, and so the motion is 
transmitted from layer to layer through the cylinder with a 


242] ORIGIN AND TRANSMISSION ‘OF SOUND. 355 


velocity that depends upon the elasticity (§ 76) of the sub- 
stance, and upon its density. This velocity is expressed by 


E 
the formula V=y/5; in which & represents the elasticity of 


the substance, and D its density ($268). Now, if we suppose 
the layer a, from any cause whatever, to execute regular vibra- 
tions, this movement will be transmitted to the succeeding 
layers with the velocity given by the formula, and, in time, 
each layer of particles in the cylinder will be executing vibra- 
tions similar to those of a. If the vibrations of a be performed 
in the time ¢, the motion will be transmitted during one com- 
plete vibration of @ to a distance s = v¢, where v is the velocity 
of propagation, say to a’, during two complete vibrations of a, 
towa distance 25 = 207, or to a’, during three complete. vibra- 
tions to a’, and soon. It is evident that the layer a’ begins 
its first vibration at the instant that @ begins its second vibra- 
tion, a” begins its first vibration at the instant that @’ begins 
its second, and a its third vibration. The layer midway be- 
tween a and a’ evidently begins its vibration just as @ com- 
pletes the first half of its vibration, and therefore moves for- 
ward while a moves backward. This condition of things exist- 
ing in the cylinder constitutes a wave motion. While a moves 
forward, the portions near it are compressed. While it moves 
backward, they are dilated. Whatever the condition at a, the 
same condition will exist at the same instant at a’, a”, etc. 
The distance aa’ = a’a”" is called a wave length, it is the dis- 
tance from any one particle to the next one of which the vibra- 
tions are in the same phase (§ Ig). . If the condition at @ and 
a’ be one of condensation, it is evident that at @, midway be- 
tween @ and a’, there must be a rarefaction. In the wave 
length aa, exist all intermediate conditions of condensation 
and rarefaction. These conditions must follow each other 
along the cylinder with the velocity of the transmitted motion, 
and they constitute a progressive wave moving with this veloc 


350 ELEMENTARY PHYSICS. [243 


ity. If the vibratory motion with which a is endowed be com- 
municated by a sounding body, the wave is a sounad-wave. If, 
instead of a cylinder of the substance, we have an indefinite 
medium in the midst of which the sounding body is placed, 
the motion is transmitted in all directions as spherical waves 
about the sounding body as a centre. 

243. Mode of Propagation of Wave Motion.—The mode 
of transmission of wave motions was first shown by Huyghens, 
and the principle involved is known as Huyghens’ principle. 
Let a (Fig. 78) be a centre from which sound originates. At 
the end of a certain time it will have reached the surface #7. 
From the preceding discussion it is evident that each particle 
of the surface sz has a vibratory motion 
similar to that at a. Any one of those par- 
ticles would, if vibrating alone, be, like a, the 
centre of a system of spherical waves, and 
each of them must, therefore, be considered 


‘\ aS a wave centre from which spherical waves. 


~< 


Nq proceed. Suppose such a wave to proceed 
from each one of them for the short dis- 
tance cd. Since the number of the element- 
ary spherical waves is very great, it is plain 
that they will coalesce to form the surface 
m'n’ which determines a new position of the 

i wave surface. In some cases the existence 

haat of these elementary waves need not be con- 
sidered, but there are many phenomena of wave motion which 
can only be studied by recognizing the fact that propagation 
always takes place as above described. 

244. Graphic Representation of Wave Motion.—In order 
to study the movements of a body in which a wave motion 
exists, especially when two or more systems of waves exist in 
the same body, it is convenient to represent the movernent 
by a sinusoidal curve, as described in § Ig. 


244] ORIGIN AND TRANSMISSION OF SOUND. 357 


Suppose the layer a (Fig. 77) to move with a simple har- 
monic motion of which the amplitude is a and the period 7, 
and let time be reckoned from the instant that the particles 
pass the position of equilibrium in a positive direction. A 
sinusoidal curve may be constructed to represent either the 
displacements of the various layers from their positions of 
equilibrium, or the velocities with which they are severally 
moving at a given time. 

To construct the first curve let the several points along OX 
{Fig. 79) represent points of the body through which the wave 


Fic. 709. 


is moving. Let Oy =a be the amplitude of vibration of each 

particle. The displacement of the particle at Oat any instant ¢ 
ae copy : 2ME 7 

after passing its position of equilibrium is y= a@ cos ee 7), 


since when ¢ is reckoned from the position of equilibrium 


Wi 
propagation of the wave, the particle at the distance x from 
the origin will have a displacement equal to that of the particle 
at Oat the instant 7, at an instant later than 7 by the time taken 


A eget h ; 
—,- Pence iia sin. lf v represent» the. velocity of 


x 
for the wave to travel over the distance %, or # seconds. Hence 
its displacement at the instant ¢ will be the same as that which 


£5 ; 
existed at O, y seconds earlier. But the displacement at O, 


x LA A. 
. seconds earlier is 


358 ELEMENTARY PHYSICS. [244 


x 
Sy suhee maee 


= asin 27 he — ee (102) 


The quantity vZ equals the distance through which the move- 
ment is transmitted during the time of one complete vibration 
of the particle at-O. Putting this equal to A, we have finally 


y= asin 27 (F — =), (103) 


Suppose ¢ = 0, and give to various values. The corresponding 
values of y will represent the displacement at that instant of the 
particle the distance of which from the origin is x Fora=o, 
y=. For*# =A, y= —a. Forxz=3h 9p =o) oe 
y=a. Forz=A,y=0, etc. Laying off these valnesor aon 
OX and erecting perpendiculars equal to the corresponding 
values of y, we have the curve Obcde .... 
The above expression for y may be put in the form 


th 
y = asin an( 7 at 
xX 


Hence, if any finite value be assigned to ¢, we shall obtain for 
y the same values as were obtained above for ¢= 0, if we in- 
tr 
ae 


crease each of the values of x by For instance, if ¢ equal 


245] ORIGIN AND TRANSMISSION OF SOUND. 359 


gu,we have y= 0 for 2 =A, y = — a for # = 4, etc.; and the 
curve becomes the dotted line 0’c’d’ .... The effect of. in- 
creasing ¢ is to displace the curve along OX in the direction of 
propagation of the wave. 

The formula for constructing the curve of velocities is derived 
in the same way as that for displacements. It is 


27a ae, 2 
J =F 608 27 bee *): (104) 


Fig. 80 shows the relation of the twocurves. The upper is the 
curve of displacement, and the lower of velocity. 


245. Composition of Wave Motions.—The composition 
of wave motions may be studied by the help of the curves ex- 
plained above. If two systems of waves coexist in the same 
body, the displacement of any particle at any instant will be 
the algebraic sum of the displacements due to the systems taken 
separately. If the curve of displacements be drawn for each 
system, the algebraic sum of the ordinates will give the ordi- 
nates of the curve representing the actual displacements. In 


360 


ELEMENTARY PHYSICS. [245 


Fig. 81 the dotted line and the light full line represent respec- 


tively the displacements due to 
two wave systems of the same 
period and amplitude. The 


/\ heavy line represents the actual 


displacement. In J the two 
systems are in the same phase; 
in II the phases differ by 4, and 
in III by 4, of a period. If both 
Wave systems move in the same 
direction, it is evident that the 
conditions of the body will be 
continuously shown by suppos- 
ing the heavy line to move in 
the same direction with the 


same-velocity. The condition represented in III is of special in- 
terest. It shows that two wave systems may completely annul 


each other. 


Aw 
Ris 


D 


Fic. 82. 


Fig. 82 represents the resultant wave when the 


periods, and consequently the wave lengths, of the two systems 


245] ORIGIN AND TRANSMISSION OF SOUND. 301 


are as 1:2. It will be noticed that the resultant curve is no 
longer a simple sinusoid. 

In the same way the resultant 
wave may be constructed for any 
number of wave systems having 
any relation of wave lengths, am- 
plitudes, and phases. <A very im- 
portant case is that of two wave sys- 
tems of the same period moving in 
opposite directions with the same 
velocity. In this case the two sys- 
tems no longer maintain the same 
relative positions, and the resultant 
curve is not displaced along the 
axis, but continually changes form. 
In Fig. 83, let the full and dotted 
lines in I represent, at a given in- 
stant, the displacements due to the 
two waves respectively. The re- 
sultant is plainly the straight line 
ab, which indicates that at that 
instant there is no displacement 
of any particle. At an instant 
later by 4 period, as shown in II, 
the wave represented by the full 
line has moved to the right $ wave 
length, while that represented by 
the dotted line has moved to the 
left the same distance. The heavy 
line indicates the corresponding 
displacements. In III, IV, V, 
etc., the conditions at instants 4, 
#, 4, etc., periods later are repre- 
sented. A comparison of these 


362 ELEMENTARY PHYSICS. (246 


figures will show that the particles at c and d are alwaysat rest, 
that the particles between c and d all move in the same direc- 
tion at the same time, and that particles on the opposite sides 
of ¢ or dare always moving in opposite directions. It follows 
that the resultant wave has no progressive motion. It is a 
stationary wave. Places where no motion occurs, such as ¢ and 
a, are called nodes. ‘The space between two nodes is an zuter- 
node or ventral segment. Vhe middle of a ventral segment, where 
the motion is greatest, is an antz-node. It will be seen later 
that all sounding bodies afford examples of stationary waves. 

246. Reflection of Waves.—When a wave reaches the 
bounding surface between two media, one of three cases may 
occur: , 

(1) The particles of the second medium may have the same 
facility for movement as those of the first. The condition at 
the boundary will then be the same as that at any point pre- 
viously traversed, and the wave will proceed as though the first 
medium were continuous. 

(2) The particles of the second medium may move with less 
facility than those of the first. Then the condensed portion of 
a wave which reaches the boundary becomes more condensed 
in consequence of the restricted forward movement of the 
bounding particles, and the rarefied portion becomes more rare- 
fied, because those particles are also restricted in their backward 
motion. The condensation and rarefaction are communicated 
backward from particle to particle of the first medium, and con- 
stitute a reflected wave. It will be-seen that, when the con- 
densed portion of the wave, in which the particles have a for- 
ward movement, reaches the boundary, the effect is a greater 
condensation, that is, the same effect as would be produced by 
imparting a backward movement to the bounding particles if 
no wave previously existed. In the direct rarefied portion of 
the wave the movement of the particles is backward, and the 
effect, at the boundary, of a greater rarefaction is what would 


247 | ORIGIN AND TRANSMISSION OF SOUND. 363 


be produced by a forward movement of those particles. The 
effect in this case is, therefore, to reverse the motion of the 
particles. It is called reflection with change of sign. 

(3) The particles of the second medium may move more 
freely than those of the first. In this case, when a wave in the 
first medium reaches the boundary, the bounding particles, 
instead of stopping with a displacement such as they would 
reach in the interior of the medium, move to a greater distance, 
and this movement is communicated back from particle to par- 
ticle as a reflected wave in which the motion has the same sign 
as in the direct wave. It is reflection without change of sign. 
The two latter cases'are extremely important in the study of 
the formation of stationary waves in sounding bodies. 

247. Law of Reflection.—Let us suppose a system of 
spherical waves departing from the point C (Fig. 84). Let mz 
be the intersection of one of 
the waves with the plane of the 
paper. Let AB be the trace of 
a plane smooth surface perpen- 
dicular to the plane of the 
paper, upon which the waves 
impinge. mo shows the position 
which the wave of which wz 1s 
a part would have occupied 
had it not been intercepted by 
the surface. From the last 
section it appears that reflection 
will take place as the wave mano 
strikes the various points of AB. In § 243 it was seen that 
any point of a wave may be considered as the centre of a 
wave system, and we may therefore take ny 7, etce the points 
of intersection of the surface AB with the wave mz when it 
occupied the positions #z’n', m''n'’, etc., as the centres of sys- 
tems of spherical waves, the resultant of which weuld be the 


Fic. 84. 


364 ELEMENTARY PHYSICS. [247 


actual wave proceeding from AL. With wz’ asa centre describe 
a sphere tangent to muoato. It is evident that this will repre- 
sent the elementary spherical wave of which the centre is 2’ 
when the main wave is at wz. Describe similar spheres with 
2’, n’’, etc., as centres. The surface zp, which envelops and is 
tangent to all these spheres, represents the wave reflected from 
AB. If that part of the plane of the paper below ABS be re- 
volved about AZ as an axis until it concides with the paper 
above AB, so will coincide with sf, s’o’ with s‘7’, etc., and hence 
no with zp. But zo is a circle with C as a centre; mp is, there- 
fore, a circle of which the centre is C’, on a perpendicular to 
AB through C, and as far below AB as C is above. When, 
therefore, a wave is reflected at a plane surface, the centres of 
the incident and reflected waves are on the same line perpen 
dicular to the reflecting surface, and at equal distances from 
the surface on opposite sides. 


CHAPTER IE 


SOUNDS AND MUSIC. 
COMPARISON OF SOUNDS. 


248. Musical Tones and Noises.—The distinction be. 
tween the impressions produced by musical tones and by noises 
is familiar to all. Physically, a musical fone is a sound the 
vibrations of which are regularand periodic. A mozseisasound 
the vibrations of which are very irregular. It may result from 
a confusion of musical tones, and is not always devoid of musi- 
cal value. The sound produced by a block of wood dropped 
on the floor would not be called a musical tone, but if blocks of 
wood of proper shape and size be dropped upon the floor in 
succession, they will give the tones of the musical scale. 

Musical tones may differ from one another in pztch, depend. 
ing upon the frequency of the vibrations; in /owduess, depending 
upon the amplitude of vibration; and in gualty, depending 
upon the manner in which the vibration is executed. In regard 
to pitch, tones are distinguished as Azghk or low, acute or grave. 
In regard to loudness, they are distinguished as /oud or soft. 
The quality of musical tones enables us to distinguish the tones 
of different instruments even when sounding the same notes. 

249. Methods of Determining the Number of Vibra- 
tions of a Musical Tone.—That the pitch of a tone depends 
upon the frequency of vibrations may be simply shown by hold- 
ing the corner of a card against the teeth of a revolving wheel. 
With a very slow motion the card snaps from tooth to tooth, 
making a succession of distinct taps, which, when the revolutions 


366 ELEMENTARY PHYSICS. [249 


ee a ae a 


are sufficiently rapid, blend together and produce a continuous 
tone, the pitch of which rises and falls with the changes of speed, 
Savart made use of such a wheel to determine the number of 
vibrations corresponding to a tone of given pitch. After regu- 
lating the speed of rotation until the given pitch was reached, 
the number of revolutions per second was determined by a 
simple attachment; this number multiplied by the number of 
teeth in the wheel gave the number of vibrations per second. 

The szvex is an instrument for producing musical tones by 
puffs of air succeeding each other at short equal intervals. A 
circular disk having in it a series of equidistant holes arranged 
in a circle around its axis is supported ‘so as to revolve parallel 
to and almost touching a metal plate in which is a similar series 
of holes. ‘The plate forms one side of a small chamber, to which 
air is supplied from an organ bellows. If there be twenty holes 
in the disk, and if it be placed so that these holes correspond 
to those in the plate, air will escape through all of them. If 
the disk be turned through a small angle, the holes in the plate 
will be covered and the escape of air will cease. If the disk be 
turned still further, at one twentieth of a revolution from its 
first position, air will again escape, and if it rotate continuously, 
air will escape twenty times in a revolution. When the rota- 
tion is sufficiently rapid, a continuous tone is produced the pitch 
of which rises as the speed increases. ‘The siren may be used 
exactly as the toothed wheel to determine the number of vibra- 
tions corresponding to any tone. 

By drilling the holes in the plate. obliquely forward in the 
direction of rotation, and those in the disk obliquely backward, 
the escaping air will cause the disk to rotate, and the speed of 
rotation may be controlled by controlling the pressure of air in 
the chamber. 

Sirens are sometimes made with several series of holes in 
the disk. ‘These serve not only the purposes described above, 


250] SOUNDS AND MUSIC. 367 


but also to compare tones of which the vibration numbers have 
certain ratios. 

The number of, vibrations of a 
sounding body may sometimes be de- 
termined by attaching to it a light 
stylus which is made to trace a curve 
upon a smoked glass or cylinder. In- 
stead of attaching the stylus to the 
sounding body directly, which is prac- 
ticable only in a few cases, it may be at- 
tached toa membrane which is caused 
to vibrate by the sound-waves which 
the body generates. A membrane re- 
produces very faithfully all the charac- 
teristics of the sound-waves, and the 
curve traced by the stylus attached to 
it gives information, therefore, not 
only in regard to the number of vibra- 
tions, but to some extent in regard 
to their amplitude and form. 


NAAN 


PHYSICAL THEORY OF MUSIC. 


250. Concord and _  Discord.— 
When two or more tones are sounded 
together, if the effect be pleasing there 
is said to be concord, if harsh, azscord. 
To understand the cause of discord, 
suppose two tones of nearly the same 
pitch to be sounded together. The re- 
sultant curve, constructed as in § 245, 
is like those in Fig. 85, which repre- 
senf the resultants when the periods 
of the components have the ratio 81 : 80 and when they have 


Mo 


Wri 


16:15 


368 ELEMENTARY PHYSICS. [25r 


the ratio 16:15. The figure indicates, what experiment veri- 
fies, that the resultant sound suffers periodic variations in in- 
tensity. When these variations occur at such intervals as to 
be readily distinguished, they are called deats. These beats 
occur more and more frequently as the numbers expressing 
the ratio of the vibrations reduced to its lowest terms become 
smaller, until they are no longer distinguishable as separate 
beats, but appear as an unpleasant roughness in the sound. 
If the terms of the ratio become smaller still, the roughness. 
diminishes, and when the ratio is £ the effect is no longer 
unpleasant. This, and ratios expressed by smaller numbers, 
as #, 8, 4, 3, 2, represent concordant combinations. 

251. Major and Minor Triads.—Three tones of which the 
vibration numbers are as 4:5:6 form a concordant combination 
called the major triad. The ratio 10:12:15 represents another 
concordant combination called the mznor triad. Fig. 86 shows 
the resultant curves for the two triads. 


ppv 
10:12:15 


Fic. 86. 


252. Intervals.—The zxzzerval between two tones is expressed 
by the ratio of their vibration numbers, using the larger as the 
numerator. Certain intervals have received names derived 
from the relative positions of the two tones in the musical 
scale, as described below. The interval 2 is called an octave; 
8, a fiftl; 4,a fourth; 4,a major third, $, a minor third. 

253. Musical Scales.—A musical scale is a series of tones 
which have been chosen to meet the demands of musical com- 


position. There are at present two principal scales in use, each 


253] SOUNDS AND MUSIC. 369 


consisting of seven notes, with their octaves, chosen with refer- 
ence to their fitness to produce pleasing effects when used in 
combination. In one, called the mayor scale, the first, third, 
and fifth, the fourth, sixth, and eighth, and the fifth, seventh, 
and ninth tones, form major triads. In the other, called the 
minor scale, the same tones form minor triads. From this it is 
easy to deduce the following relations: 


MAJOR SCALE, 


7 nal 
Tone Number...........005. I 2 3 4 5 6 7 8 9 
“Mio sho pl 45 Sk Reeves Bae Bab GT RA BS nh Ty 
ORES Paige) Ss nik © 5b sie ole 30's Goren wires mifa sol. ti Ja: 2) sia \ uti)» Fe 
Number of vibrations........ m gm §m #m §m §m 15m 2m 2m 

9 10 1 
Intervals from tone to tone.. ef te oe 48 eae 
MINOR SCALE, 

MC OMUMDES, 5s o.¢es ccc twees I 2 3 4 5 6 2 8 9 
he pics sds ncis 54 slo a eh RES? sola 2) OD EO ORM Chel NN CA He Tel 2) 
BME alex ss se sr eehc es iseeer tut, reli imi. fa) sol ia si 
Number of vibrations........ m gm gm $m gm gm gm 2m 3m 
Intervals from tone to tone.. $ ig 10 3 28 g 640 


The derivation of the names of the intervals will now be 
apparent. For example, an interval of a third is the interval 
between any tone of the scale and the third one from it, count- 
ing the first as I. If we consider the intervals from tone to 
tone, it is seen that the pitch does not rise by equal steps, but 
that there are three different intervals, $, 49, and 1%. The first 
two are usually considered the same, and are called whole tones. 
The third is a half-tone or semttone. 

It is desirable to be able to use any tone of a musical in- 
strument as the first tone or ¢ouzc of a musical scale. To per- 
mit this, when the tones of the instrument are fixed, it is plain 
that extra tones, other than those of the simple scale, must be 
provided in order that the proper sequence of intervals may be 
maintained. Suppose the tonic to be transposed from C to D. 

24 


370 ELEMENTARY PHYSICS. [253 


The semitones should now come, in the major scale, between F 
and G, and C’ and D’, instead of between E and F, and B and 
C’. To accomplish this, a tone must be substituted for F and 
another for C’.. These are called F sharp and C’ sharp respec- 
tively, and their vibration numbers are determined by multiply- 
ing the vibration numbers of the tones which they replace by 34. 
The introduction of five such extra tones, making twelve in 
the octave, enables us to preserve the proper sequence of whole 
tones and semitones, whatever tone is taken as the tonic. But 
if we consider that the whole tones are not all the same, and 
propose to preserve exactly all the intervals of the transposed 
scale, the problem becomes much more difficult, and can only 
be solved at the expense of too great complication in the in- 
strument. Instead of attempting it, a system of tuning, called 
temperament, is used by which the twelve tones referred to above © 
are made to serve for the several scales, so that while none are 
perfect, the imperfections are nowhere marked. The system of 
temperament usually employed, or at least aimed at, called the 
even temperament, divides the octave into twelve equal semi- 
tones, and each interval is therefore the twelfth root of 2. 
With instruments in which the tones are not fixed, like the 
violin for instance, the skilful performer may give them their 
exact value. 

For convenience in the practice of music and in the con- 
struction of musical instruments, a standard pitch must be 
adopted. This pitch is usually determined by assigning a fixed 
vibration number to the tone above the middle C of the piano, 
“represented by the letter A’. This number is about 440, but 
varies somewhat in different countries and at different times. 
In the instruments made by Konig for scientific purposes, the 
vibration number 256 is assigned to the middle C. This has 
the advantage that the vibration numbers of the successive 
octaves of this tone are powers of 2. 


Sle eg ad 0 el cl AO 
VIBRATIONS OF SOUNDING BODIES. 


254. General Considerations.—The principles developed 
in § 246 apply directly in the study of the vibrations of sound- 
ing bodies. When any part of a body which is capable of act- 
ing as a sounding body is set in vibration, a wave is propagated 
through it to its boundaries, and is there reflected. The re- 
flected wave, travelling away from the boundary, in conjunction 
with the direct wave going toward it, produces a stationary 
wave. These stationary waves are characteristic of the motion 
of allsounding bodies. Fixed points of a body often determine 
the position of nodes, and in all cases the length of the wave 
must have some relation to the dimensions of the body. 

255. Organ Pipes.—A column of air, enclosed in a tube of 
suitable dimensions, may be made to vibrate and become a 
sounding body. Let us suppose a tube closed at one end and 
open at the other. If the air particles at the open end be sud- 
denly moved inward, a pulse travels to the closed end, and is 
there reflected with change of sign ($§ 246). It returns to the 
open end and is again reflected, this time without change of 
sign, because there is greater freedom of motion without than 
within the tube. As it starts again toward the closed end, the 
air particles that compose it move outward instead of inward. 
If they now receive an independent impulse outward, the two 
effects are added and a greater disturbance results. So, by 
properly timing small impulses at the open end of the tube, the 
air in it may be made to vibrate strongly. 


372 ELEMENTARY PHYSICS. [255. 


If a continuous vibration be maintained at the open end of 
the tube, waves follow each other up the tube, are reflected with 
change of sign at the closed end, and returning, are reflected 
without change of sign at the open end. Any given wave a, 
therefore, starts up the tube the second time with its phase 
changed by half a period. The direct wave that starts up 
the tube at the same instant must be in the same phase as the 
reflected wave, and it therefore differs in phase half a period 
from the direct wave a. In other words, any wave returning 
to the mouth-piece must find the vibrations there opposite in 
phase to those which existed when it left. This is possible only 
when the vibrating body makes, during the time the wave is 
going up the tube and back, I, 3, 5, or some odd number of 
half-vibrations. By constructing the curves representing the 
stationary wave resulting from the superposition of the two 
systems of vibrations, it will be seen that there is always a node 
at the closed end of the tube and an anti-node at the mouth. 
When there is 1 half-vibration while the wave travels up and 
back, the length of the tube is 4 the wave length; when there 
are 3 half-vibrations in the same time, the length of the tube is 
# the wave length, and there is a node at one third the length 
of the tube from the mouth. 

If the tube be open at both ends, reflection without change 
of sign takes place in both cases, and the reflected wave starts 
up the tube the second time in the same phase as at first, The 
vibrations must therefore be so timed that I, 2, 3, 4, or some 
whole number of complete vibrations are performed while the 
wave travels up the tube and back. A construction of the 
curve representing the stationary wave in this case will show, 
for the smallest number of vibrations, a node in the middle of 
the tube and an anti-node at each end. The length of the 
tube is therefore 4 the wave length for this rate of vibration. 
The vibration numbers of the several tones produced by an 
open tube are evidently in the ratio of the series of whole num- 


256] VIBRATIONS OF SOUNDING BODIES. an 


bers I, 2, 3, 4, etc., while for the closed tube only those tones 
can be produced of which the vibration numbers are in the ratio 
of the series of odd numbers 1, 3, 5, etc. It is evident also that 
the lowest tone of the closed tube is an octave lower than that 
of the open tube. 

This lowest tone of the tube is called the fundamental, and 
the others are called overtones, or harmonics. These simple 
relations between the length of the tube and length of the wave 
are only realized when the tubes are so narrow that the air 
particles lying in a plane cross-section are all actuated by the 
same movement. This is never the case at the open end of the 
tube, and the distance from this end to the first node is, there- 
fore, always less than a quarter wave length. 

256. Modes of Exciting Vibrations in Tubes.—lIf a tun- 
ing fork be held in front of the open mouth of a tube of proper 
length, the sound of the fork is strongly reinforced by the 
vibration of the air in the tube. If we merely 
blow across the open end of a tube, the agitation 
of the air may, by the reaction of the returning 
reflected pulses, be made to assume a regular vi- 
bration of the proper rate and the column made 
tosound. In organ pipes a mouthpiece of the 
form shown in Fig. 87 is often em- 
ployed. The thin sheet of air projected 
against the thin edge is thrown into 
vibration. Those elements of this vi- 
bration which correspond in frequency 
with the pitch of the pipe are strongly 
reinforced by the action of the station- 
ary wave set up in the pipe, and hence 
the tone proper to the pipe is produced. 
Sometimes veeds are used, as shown in Fig. 87a. The air es. 
caping from the chamber @ through the passage ¢ causes the 
reed 7 to vibrate. This alternately closes and opens the passage, 


Fic, 87. Fic. 87a. 


374 ELEMENTARY PHYSICS. [257 


and so throws into vibration the air in the pipe. If the reed 
be stiff, and have a determined period of vibration of its own, 
it must be tuned to suit the period of the air column which it 
is intended to set in vibration. If the reed be very flexible it 
will accommodate itself to the rate of vibration of the air col- 
umn, and may then serve to produce various tones, as in the 
clarionet. 

In instruments like the cornet and bugle, the lips of the 
player act as a reed, and the player may at will produce many 
of the different overtones. In that way melodies may be played 
without the use of keys or other devices for changing the length 
of the air column. 

Vibrations may be excited in a tube by placing a gas flame 
at the proper point init. The flame thus employed is called a 
singing flame. ‘The organ of the voice is a kind of reed pipe 
in which little folds of membrane, called vocal chords, serve as. 
reeds which can be tuned to different pitches by muscular 
effort, and the cavity of the mouth and larynx serves as a pipe 
in which the mass of air may also be changed at will, in form 
and volume. 

257. Longitudinal Vibrations of Rods.—A rod free at both 
ends vibrates as the column of air in an open tube. Any dis- 
placement produced at one end is transmitted with the velocity 
of sound in the material to the other end, is there reflected with- 
out change of sign and returns to the starting point to be re- 
flected again exactly as in the open tube. The fundamental 
tone corresponds to a stationary wave having a node at the cen- 
tre of the rod. ; 

258. Longitudinal Vibrations of Cords.—Cords fixed at 
both ends may be made to vibrate by rubbing them lengthwise. 
Here reflection with change of sign takes place at both. ends, 
which brings the wave as it leaves the starting point the second 
time to the same phase as when it first left it, and there must 
be, therefore, as in the open tube, I, 2, 3, 4, étc., vibrations 


259] VIBRATIONS OF SOUNDING BODIES. 375 


while the wave travels twice the length of the cord. The veloc- 
ity of transmission of a longitudinal displacement in a wire de- 
pends upon the elasticity and density of the material only. 
The velocity and the rate of vibration are, therefore, nearly 
independent of the stretching force. 

259. Transverse Vibrations of Cords.—lIf a transverse 
vibration be given to a point upon a wire fastened at both ends, 
everything relating to the reflection of the wave motion and 
the formation of stationary waves is the same as for longitudinal 
displacements. The velocity of transmission, and consequently 
the frequency of the vibrations, are, however, very different. 
If the cord offer no resistance to flexure, the force tending to 
restore it to its position of equilibrium is entirely due to the 
stretching force. This, therefore, takes the place of the elas- 
ticity in the formula for transmission of longitudinal vibrations 
(§ 268). The mass of the cord per unit length takes the place 
of the density in the same formula. Thus we have the formula 


for the velocity 
= Vie 
m 


where P is the stretching force and m the mass per unit length. 
The greatest time of vibration, the time required for the wave 
to travel twice the length of the string, is 


l= 2 eh Ly /"% (105) 


and the number of vibrations per second is 


T= 2 I P 
— ot as 6 
A i s1\/ 5 (10 ) 


376 ELEMENTARY PHYSICS. [260 


Hence, the number of vibrations of a string is inversely as the 
length, directly as the square root of the tension, and inversely 
as the square root of the mass per unit length. These laws are 
readily verified by experiment. 

260. Transverse Vibrations of Rods, Plates, etc.—The 
vibrations of rods, plates,and bells are all cases of stationary 
waves resulting from systems of waves travelling in opposite di-. 
rections. Subdivision into segments occurs, but, in these cases, 
the relations of the various overtones are not so simple as in 
the cases before considered. For a rod fixed at one end, sound- 
ing its fundamental tone, there is a node at the fixed end only. 
For the first overtone there is a second node near the free end 
of the rod, and the number of vibrations is a little more than 
six times the number for the fundamental. 

A rod free at both ends has two nodes when sounding its 
fundamental, as shown in Fig.88. The distance of these nodes 

_ from the ends is about 2 the length of 

— =<. the rod. If the rod be bent, themodges 

Fic. 88. approach the centre until, when it has 

assumed the U form like a tuning-fork, the two nodes are very 
near the centre. This will be understood from Fig. 89. 


Fic. 89. 


The nodal lines on plates may be shown by fixing the plate 
in a horizontal position and sprinkling sand over its surface. 
When the plate is made to vibrate, the sand gathers at the nodes 


261] VIBRATIONS OF SOUNDING BODIES. ive 


and marks their position. The figures thus formed are known 
as Chladni’s figures. 

261. Communication of Vibrations.—If several pendulums 
be suspended from the same support, and one of them be made 
to vibrate, any others which have the same period of vibration 
will soon be found in motion, while those which have a different 
period will show no signs of disturbance. The vibration of the 
first pendulum produces a slight movement of the support which 
is communicated alike to all the other pendulums. Each move- 
ment may be considered as a slight impulse, which imparts to 
each pendulum a very small vibratory motion. For those pen- 
dulums having the same period as the one in vibration, these 
impulses come just in time to increase the motion already pro- 
duced, and so, after a time, produce a sensible motion ; while for 
those pendulums having a different period, the vibration at first 
imparted will not keep time with the impulses, and these will 
therefore as often tend to destroy as to increase the motion. 
It is important to note that the pendulum imparting the motion 
loses all it imparts. This is not only true of pendulums, but of 
all vibrating bodies. Two strings stretched from the same sup- 
port and tuned to unison will both vibrate when either one is 
caused to sound. A tuning-fork suitably mounted on a sound- 
ing-box will communicate its vibrations to another tuned to 
exact unison even when they are thirty or forty feet apart and 
only airintervenes. In this case it isthe sound-wave generated 
by the first fork which excites the second fork, and in so doing 
the wave loses a part of its own motion, so that beyond the 
second fork, on the line joining the two, the sound will be less 
intense than at the same distance in other directions. 

Air columns of suitable dimensions will vibrate in sympathy 
with other sounding bodies. If water be gradually poured into 
a deep jar, over the mouth of which is a vibrating tuning-fork, 
there will be found in general acertain length of the air column 
for which the tone of the fork is strongly reinforced. From 


378 ELEMENTARY PHYSICS. [26r 


the theory of organ pipes, it is plain that this length corresponds 
approximately to a quarter wave length for that tone. In this 
case, also, when the strongest reinforcement occurs, the sound of 
the fork will rapidly die away. The sounding-boxes on which 
the tuning-forks made by Ké6nig are mounted are of such 
dimensions that the enclosed body of air will vibrate in unison 
with the fork, but they are purposely made not quite of the 
dimensions for the best resonance, in order that the forks may 
not too quickly be brought to rest. 

A membrane or a disk, fastened by its edges, may respond 
toand reproduce more or less faithfully a great variety of sounds. 
Hence such disks, or diaphragms, are used in instruments like 
the telephone and phonograph, designed to reproduce the 
sounds of the voice. The phonograph consists of a mouthpiece 
and disk similar to that used in the telephone, but the disk 
has fastened to its centre, on the side opposite the mouthpiece, 
a short stiff stylus, which serves to record the vibrations of the 
disk upon a sheet of tinfoil or wax moved along beneath it. 
The foil is wrapped upon a cylinder having a spiral groove on 
its surface, and upon its axle a screw thread of the same pitch 
works in a fixed nut so that, when the cylinder revolves, it has 
also an endwise motion, such that a fixed point would follow 
the spiral groove on itssurface. ‘To use the instrument, the disk 
is placed in position with the stylus attached adjusted to enter 
the groove in the cylinder and slightly indenting the foil. The 
cylinder is revolved while sounds are produced in front of the 
disk. The disk vibrates, causing the stylus to indent the foil 
more or less deeply, so leaving a permanent record. If now the 
cylinder be turned back to the starting-point and then turned 
forward, causing the stylus to go over again the same path, the 
indentations previously made in the foil now cause the stylus, 
and consequently the disk, to vibrate and reproduce the sound 
that produced the record. 

The sounding-boards of the various stringed instruments are 


261] VIBRATIONS OF SOUNDING BODIES. 379 


in effect thin disks, and afford examples of the reinforcement 
of vibrations of widely different pitch and quality by the same 
body. The strings of an instrument are of themselves insuffi- 
cient to communicate to the air their vibrations, and it is only 
through the sounding-board that the vibrations of the string 
can give rise to audible sounds. The quality of stringed instru- 
ments, therefore, depends largely upon the character of the 
sounding-board. 

The tympanum of the ear furnishes another example of the 
facility with which membranes respond to a great variety of 
sounds, 


CHAPTER Lv. 
ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 


262. Quality.—As has already been stated, the tones of di - 
ferent instruments, although of the same pitch and intensity 
are distinguished by their gualty. It was also stated that the 
quality of a tone depends upon the manner in which the vibra- 
tion is executed. The meaning of this statement can best be 
understood by considering the curves which represent the 


FIG. go. 


vibrations. In Fig. 90 are given several forms of vibration 
curves of the same period. 

Every continuous musical tone must result from a periodic 
vibration, that is, a vibration which, however complicated it 
may be, repeats itself at least as frequently as do the vibrations 
of the lowest audible tone. According to Fourier’s theorem 
(§ 19), every periodic vibration is resolvable into simple har- 
monic vibrations having commensurable periods. It has been 


262] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 381 


seen that all sounding bodies may subdivide into segments, and 
produce a series of tones of which the vibration periods gener- 
ally bear a simple relation to one another. These may be pro- 
duced simultaneously by the same body, and so give rise to 
complex tones the character of which will vary with the nature 
and intensity of the simple tones produced. It has been held 
that the quality of a complex tone is not affected by change of 
phase of the component simple tones relative to each other. 
Some experiments by Kénig seem to indicate, however, that 
the quality does change when there is merely change of phase. 


Fic. or. 


In Fig. 91 are shown three curves, each representing a fun- 
damental accompanied by the harmonics uptothetenth. The 


Me itineec!. (ab owk 


Fic. 92. 


curves differ only in the different phases of the components 
relative to each other. 


382 ELEMENTARY PHYSICS. [263 


Fig. 92 shows similar curves produced by a fundamental 
accompanied by the odd harmonics. 

263. Resonators for the Study of Complex Tones.—An 
apparatus devised by Helmholtz serves to analyze complex 
tones and indicate the simple tones of which they are composed. 
It consists of a series of hollow spheres or cylinders, called 
vesonators, which are tuned to certain tones. If a tube lead 
from the resonator to the ear and a sound be produced, one of 


0 


A 


FIG. 93. 


the components of which is the tone to which the resonator is 
tuned, the mass of air init will be set in vibration and that tone 
will be clearly heard; or, if the resonator be connected bya 
rubber tube to a manometric capsule (§ 241), the gas flame con- 
nected with the capsule will be disturbed whenever the tone to 


265] ANALYSIS OF SOUNDS AND SOUND SENSATIONS. 383 


which the resonator is tuned is produced in the vicinity, either 
by itself or as acomponent of a complex tone. By trying the 
resonators of a series, one after another, the several compo- 
nents of a complex tone may be detected and its composition 
demonstrated. 

264. Vowel Sounds.—Helmholtz has shown that the dif- 
ferences between the vowel sounds are only differences of 
quality. That the vowel sounds correspond to distinct forms 
of vibration is well shown by means of the manometric flame. 
By connecting a mouthpiece to the rear of the capsule, and 
singing into it the different vowel sounds, the flame images 
assume distinct forms for each. Some of these forms are 
shown in Fig. 93. 

265. Optical Method of Studying Vibrations. — The vi- 
bratory motion of sounding bodies may sometimes be studied 


OS) A BLO 
OPLTOS(\o4 


Fic. 94. 


to advantage by observing the lines traced by luminous points 
upon the vibrating body or by observing the movement of a 
beam of light reflected from a mirror attached to the body. 

Young studied the vibrations of strings by placing the 
string where a thin sheet of light would fall across it, so as to 
illuminate a single point. When the string was caused to 
vibrate, the path of the point appeared as a continuous line, in 
consequence of the persistence of vision. Some of the results 
which he obtained are given in Fig. 94, taken from Tyndall on 
Sound. 


384 ELEMENTARY PHYSICS. [265 


The most interesting application of this method was made 
by Lissajous to illustrate the composition of vibratory motions 
at right angles to each other. If a beam of light be reflected 
to a screen from a mirror attached to a tuning-fork, when the 
tuning-fork vibrates the spot on the screen will describe a sim- 
ple harmonic motion and will appear as a straight line of light. 
If the beam, instead of being reflected to a screen, fall upon a 
mirror attached to a second fork, mounted so as to vibrate in 


a plane at right angles to the first, the spot of light will, when 
both forks vibrate, be actuated by two simple harmonic mo- 
tions at right angles to each other and the resultant path will — 
appear as a curve more or less complicated, depending upon 
the relation of the two forks to each other as to both period 
and phase ($ 19). Fig. 95 shows some of the simpler forms of 
these curves. The figures of the upper line are those produced 
by two forks in unison; those of the second line by two forks 
of which the vibration numbers are as 2:1; those of the lower 
line by two forks of which the vibration numbers are as 3: 2. 


ihe Ya ai 4 SEN 
EFFECTS OF THE COEXISTENCE OF SOUNDS. 


266. Beats.—It has already been explained ($250) that, 
when two tones of nearly the same pitch are sounded together, 
variations of intensity, called deats, are heard. Helmholtz’s 
theory of the perception of beats was, that, of the little fibres 
in the ear which are tuned so as to vibrate with the various 
tones, those which are nearly in unison affect one another so as 
to increase and diminish one another’s motions, and hence that 
no beats could be perceived unless the tones were nearly in 
unison. Beats are, however, heard when a tone and its octave 
are not quite in tune, and, in general, a tone making z vibra- 
tions produces zz beats when sounded with a tone making 
2nu + m, 3n + m, etc., vibrations. This was explained in ac- 
cordance with Helmholtz’s theory, by assuming that one of the 
harmonics of the lower tone, which is nearly in unison with 


Fic. 96. 


the upper, causes the beats, or, in cases where this is inad- 
missible, that they are caused by the lower tone in conjunc- 
tion with a resultant tone (§ 267). An exhaustive research by 


K6nig, however, has demonstrated that beats are perceived 
25 


386 ELEMENTARY PHYSICS. [266 


when neither of the above suppositions is admissible. Figs. 
96 and’ 97 show that the resultant vibrations are affected by 
changes of amplitude similar to, though less in extent than, 
the changes which occur when the tones are nearly in unison. 
In Fig. 96, I represents a flame image obtained when two tones 
making z and z + m vibrations respectively, are produced to- 


15:16 
16 
15:29 
| 
II 
15:31 
III 
15:46 


Fic. 97. 


gether, and II represents the image when the number of 
vibrations are z and 2x+m. Fig.:97 shows traces obtained 
mechanically. In I the numbers of the component vibrations 
were zand2-—+ m, in II and II] z and 2z + m, and in 1V z 
and 3z-+-m. In all these cases a variation of amplitude occurs 
during the same intervals, and it seems reasonable to suppose 
that those variations of amplitude should cause variations in 
intensity in the sound perceived. 


267] BPREGTS OF THE COEXISTENCE OF SOUNDS. 387 


SSeS 


Cross has shown that the beating of two tones is perfectly 
well perceived when the tones themselves are heard separately 
by the two ears; one tone being heard directly by one ear, 
while the other, produced in a distant room, is heard by the 
other ear by means of a telephone. Beats are also perceived 
when tones are produced at a distance from each other and from 
the listener, who hears them by means of separate telephones 
through separate lines. In this case there is no possibility of 
the formation of a resultant wave, or of any combination of the 
two sounds in the ear. 

267. Resultant Tones.—Resultant tones are produced by 
combinations of two tones. Those most generally recognized 
have a vibration number equal to the sum or difference of the 
vibration numbers of their primaries. For instance, ut,, making 
2048 vibrations, and re,, making 2304 vibrations, when sounded 
together give ut,, making 256 vibrations. These tones are 
only heard well when the primaries are loud, and it requires an 
effort of the attention and some experience to hear them at all. 
Summation tones are more difficult to recognize than differ- 
ence tones, nevertheless they have an influence in determining 
the general effect produced when musical tones are sounded 
together. Other resultant tones may be heard under favorable 
conditions. As described above, two tones making z anda m 
vibrations respectively, when wz is considerably less than z, give 
a resultant tone making mw vibrations, but a tone making 7 
vibrations in combination with one making 2”-+- m, 3u + m, 
or +u-+ m vibrations, gives the same resultant. This has 
sometimes been explained by assuming that intermediate re- 
sultants are produced, which, with one of the primaries, pro- 
duce resultants of a higher order. In the case of the two tones 
making z and 32-+ m vibrations, for instance, the first differ- 
ence tone would make 2z + m vibrations. This tone and the 
one making z# vibrations would give the tone making z+ m 
vibrations; this tone, in turn, and the one making z vibrations 


388 ELEMENTARY PHYSICS. [267 
would give the tone making m vibrations. This last tone is. 
the one which is heard most plainly, and it seems difficult to 
admit that it can be the resultant of tones which are only heard 
very feebly, and often not at all. In Fig. 97 are represented 
the resultant curves produced in several of these cases. The 
first curve corresponds to two tones of which the vibration 
numbers are as 15:16. It shows the periodic increase and de- 
crease in amplitude, occurring once every 15 vibrations, which, 
if not too frequent, give rise to beats ($250). If the pitch of 
the primaries be raised, preserving the relation 15:16, the 
beats become more frequent, and finally a distinct tone is. 
heard, the vibration number of which corresponds to the num- 
ber of beats that should exist. It was for a long time consid- 
ered that the resultant tone was merely the rapid recurrence of 
beats. Helmholtz has shown by a mathematical investigation 
that a distinct’ wave making zz vibrations will result from the 
coexistence of two waves making z and x-+ wm vibrations, and . 
he believes that mere alternations of intensity, such as consti- 
tute beats, occurring ever so rapidly cannot produce a tone. 

In II and III (Fig. 97) are the curves resulting from two. 
tones, the intervals between which are respectively 


15:200= 2X 15—1) and 15:31(—2 XK a5 


Running through these may be seen a periodic change corre- 
sponding exactly in period to that shown in I. The same is 
true also of the curve in IV, which is the resultant for two 
tones the interval between which is 15:46(= 3 X 15 +1). In 
all these cases, as has been already said (§ 266), if the pitch of 
the components be not too high, one beat is heard for every 15 
vibrations of the lower component. Fig. 96 shows the flame 
images for the intervals ~:2x-+ m and ~:2nu-++ m. The Vary- 
ing amplitudes resulting in # beats per second are very evident 
in both. In all these cases, also, as the pitch of the compo- - 


267] mi heG la Of Pile COLXISTENCE OF SOUNDS, 389 


nents rises the beats become more frequent, and finally a re- 
sultant tone is heard, having, as already stated, one vibration 
for every 15 vibrations of the lower component. In Fig. 98 


1:15:29 


1:15:29 . 
II 
Fic, 08. 


are shown two resultant curves having three components of 
which the vibration numbers are as 1:15:29. In I the three 
components all start in the same phase. In II, when 15 and 29 
are in the same phase, I is in the opposite phase. 


\ 


CHAPTER VE 
VELOCITY OF SOUND. 


268. Theoretical Velocity.—The disturbance of the parts 
of any elastic medium which is propagating sound is assumed, 
in theoretical discussions, to take place in the line of direction 
of the propagation of the sound, and to be such that the type 
of the disturbance remains unaltered during its propagation. 
The velocity of propagation of such a disturbance may be in- 
vestigated by the following method, due to Rankine. 

Let us consider, as in § 242, a portion of the elastic medium 
in the form of an indefinitely long cylinder. If a disturbance 
be set up at any cross-section of this cylinder (Fig. 99), which 
consists of a displacement of the matter in that cross-section 
in the direction of the axis of the cylinder, it will, by hy- 
pothesis, be propagated in the direction of the axis with a con- 
stant velocity VY, which is to be determined. If we consider 
any cross-section of the cylinder which is traversed by the dis- 
turbance, the matter which passes through it at any instant will 


B A 


FIG. 99. 


have a velocity which may vary from zero to the maximum 
velocity of the vibrating matter, either positively when this ve- 
locity is in the direction of propagation of the disturbance, or 
negatively when it is opposite to it. 

If we now conceive an imaginary cross-section A to move 


268] VELOCITY OF SOUND. 391 


along the cylinder with the disturbance with the velocity V, 
the velocity of the particles in it at any instant will be always 
the same. Let us call this velocity v,. The velocity of the 
cross-section relative to the moving particles in it is then 
V—v,. If we represent by d, the density of the medium at 
the cross-section through which the velocity of the particles 
is Y,, which is the same for all positions of the moving cross- 
section, and if we assume that the area of the cross-section is 
unity, then the quantity of matter J7 which passes through 
the moving cross-section in unit time is 


M=d{V — 2). 


If we conceive any other cross-section B to be moving with 
the disturbance in a similar manner, the same quantity of mat- 
ter 7 will pass through it in unit time, since the two cross- 
sections move with the same velocity and the density of the 
matter between them remains the same. Hence we have 
M = ad; (V — v5), where d; and v, represent the quantities at the 
cross-section 4 corresponding to those at the cross-section A 
represented by d@, and v,. Hence d,V—v,.) =adi{V — %). 
Since this equation is true whatever be the distance between 
the cross-sections, it is true for that position of the cross-section 
B for which v; = 0, and for which d;= WD, the density of 
the medium in its undisturbed condition. Hence we have 


M= DV, a,{V — vz) = DV, and 


VU, @a—D 
so Sena (107) 


If the disturbance be small, the expression on the right is 
approximately the condensation per unit volume of the me- 
dium at the cross-section A, and the equation shows that the 
1atio of the velocity of the matter passing through the cross- 


392 ELEMENTARY PHYSICS. [269 


section A to the velocity of propagation of the disturbance is 
equal to the condensation at that cross-section. 

Now, to eliminate the unknown expressions v, and d,, we 
must find a new equation involving them. A quantity of mat- 
ter M/ enters the region between the two moving cross-sections 
with the velocity v,, and an equal quantity leaves the region 
with the velocity v;. The difference of the momenta of the 
entering and outgoing quantities is W7(v,—v,). This differ- 
ence can only be due to the different pressures fg, and pf; on the 
moving cross-sections, since the interactions of the portion of 
matter between those cross-sections cannot change the momen- 
tum of that portion. Hence we have 


M (Ve — Us) = pa — Po 


If we for convenience assume v%; = 0, we have pf; = P, the 
pressure in the medium in its undisturbed condition. If we 


further substitute for v, its value, we obtain WV = d, & — 


If the Pao in pressure and density be small, the quantity 


Lie 
da dad, — D 
If we further substitute for JZ its value VD, we obtain finally 


pode E 
V mare v=/%. (108) 


269. Velocity of Sound in Air.—In air at constant tem- 
perature the elasticity is numerically equal to the pressure 
(§ 77). The compressions and rarefactions in a sound-wave 
occur so rapidly that during the passage of a wave there is no 
time for the transfer of heat, and the elasticity to be considered, 
therefore, is the elasticity when no heat enters or escapes 


($ 158). 


uae £, the modulus of elasticity of the medium. 


269] VELOCITY OF SOUND. 303 


If the ratio of the two elasticities be represented by y we 
have for the elasticity when no heat enters or escapes & = yP, 
and the velocity of a sound-wave in air at zero temperature is 
given by 


The coefficient vy equals 1.41. VP is the pressure exerted by a 
column of mercury 76 centimetres high and with a cross-section 
of one square centimetre, or 76 X 13.59 X 981 = 1013373 dynes 
per square centimetre. D equals 0.001293 grams at 0°, hence 


v4 1A! oui BEGAN 


or 332.4 metres per second. 

Since the density of air changes with the temperature, the 
velocity of sound must also change. If d, represent the den- 
sity at temperature ¢, and d@, the density at zero, 


« a, 
ae tek? 


from § 128. The formula for velocity then becomes 
Vee et _ zt). 


This formula shows that the velocity at any temperature is the 
velocity at 0° multiplied by the square root of the factor of ex- 
pansion. 


394 ELEMENTARY PHYSICS. [270 


270. Measurements of the Velocity of Sound.—The ve- 
locity of sound in air has been measured by observing the time 
required for the report of a gun to travel to a known distance. 

One of the best determinations was that made in Holland 
in 1822. Guns were fired alternately at two stations about nine 
miles apart. Observers at one station observed the time of 


seeing the flash and hearing the report from the other. The 


guns being fired alternately, and the sound travelling in oppo- 
site directions, the effect of wind was eliminated in the mean 
of the results at the two stations. It is possible, by causing the 
sound-wave to act upon diaphragms, to make it record its own 
time of departure and arrival, and by making use of some of the 
methods of estimating very small intervals of time the velocity 
of sound may be measured by experiments conducted within 
the limits of an ordinary building. 

The velocity of sound in water was determined on Lake 
Geneva in 1826 by an experiment analogous to that by which 
the velocity in air was determined. 

In § 255 and § 257 it is shown that the time of one vibration 
of any body vibrating longitudinally is the time required fora 
sound-wave to travel twice the distance between two nodes. 
The velocity may, therefore, be measured by determining the 
number of vibrations per second of the sound emitted, and 
measuring the distance between the nodes. 

In an open organ-pipe, or a rod free at both ends, when the 
fundamental tone is sounded the sound travels twice the length 
of the rod or pipe during the time of one complete vibration. 
If rods of different materials be cut to such lengths that they 
all give the same fundamental tone when vibrating longitudi- 
nally, the ratio of their lengths will be that of the velocity of 
sound in them. 

In Kundt’s experiment, the end of a rod having a light disk 
attached is inserted in a glass tube containing a light powder 
strewn over its inner surface. When the rod is made to vibrate 


st A 
, 8 


270] VELOCITY OF SOUND. 305 


longitudinally, the air-column in the tube, if of the proper length, 
is made to vibrate in unison with it. This agitates the powder 
and causes it to indicate the positions of the nodes in the vi- 
brating air-column. The ratio of the velocity of sound in the 
solid to that in air is thus the ratio of the length of the rod ta 
the distance between the nodes in the air-column. 


LIGHT. 


CrUAR Dt ha 
PROPAGATION OF LIGHT. 


271. Vision and Light.—The ancient philosophers, before 
Aristotle, believed that vision consisted in the contact of some 
subtle emanation from the eye with the object seen. Aristotle 
showed the absurdity of this view by suggesting that if it were 
true, one should be able to see in the dark. Since his time, it 
has been generally admitted that vision results from something 
proceeding from the body seen to the eye, and there impress- 
ing the optic nerve. This we call Zighz. 

Optics treats of the phenomena of light. It is conveniently 
divided into two branches, Physical Optics, which treats of the 
phenomena resulting from the propagation of light through 
space and through different media, and Physiological Optics, 
which treats of the sense of vision. 

272. Theories of Light.—At the time of Newton, light 
was generally considered to consist of particles which were not 
those of ordinary matter, projected from a luminous body, 
and exciting vision by their impact on the retina. This theory 
was strongly supported by Newton himself, who found in it 
plausible explanations of most luminous phenomena then 
known. But even in Newton’s time phenomena were known 
which could only be explained by assigning to the luminiferous 


274] PROPAGATION OF LIGHT. 397 


particles very improbable forms and motions, and, since his 
time, facts have been discovered that are inconsistent with any 
emission theory. 

The uwxdulatory theory, which is the one universally adopted, 
assumes that light is a wave motion in an elastic medium per- 
vading all space. All luminous bodies excite in this medium 
systems of waves which are propagated according to the same 
mechanical laws as those which govern wave systems in other 
media, some of which have been developed in § Ig and §§ 242- 
245. The undulatory theory has stood weli the test of ex- 
plaining newly discovered phenomena, and has moreover led 
to the discovery of phenomena not before known. The ob- 
jections to the theory are that it requires the hypothesis of a 
medium of the existence of which there is no direct evidence, 
pervading space, and requires us to ascribe to that medium 
properties unlike those of any body with which we are ac- 
quainted. 

A modified form of the undulatory theory, known as the 
electromagnetic theory of light, was proposed by Maxwell. It 
will be briefly presented after the facts connecting light and 
electricity have been considered. 

273. Wave Surfaces.—In § 243 is explained the genera! 
mode of propagation ‘of wave motion in accordance with 
Huyghens’ principle. When light, emanating from a point, 
proceeds with the same velocity in all directions, the wave 
fronts are evidently concentric spherical surfaces. There are, 
however, many cases, especially in crystalline bodies, of un- 
equal velocities in different directions. In these cases the 
wave fronts are not spherical but ellipsoidal, or surfaces of still 
greater complexity. 

274. Straight Lines of Light.—When a small screen y 
(Fig. 100) is placed between the eye and a luminous point, 
the luminous point is no longer visible. Light cannot reach 
the eye by the curved or broken line PAZ, and is therefore 


398 ELEMENTARY PHYSICS. [274 


said to move in straight lines. This seems not to accord with 
Huyghens’ principle which makes any wave front the resultant 
of an infinite number of elementary waves proceeding from the 


A 


E 
Ee tin ae aca 


Fic. 100. 


various points of the same wave front in one of its earlier posi- 
tions, It can, however, easily be shown that when the wave 
lengths are small, the disturbance at any point P(Fig. 1o1)is due 

almost wholly to a very small portion 


: of the approaching wave. Let us 
¢ consider first the case of an isotropic 
b medium, in which light moves in all 
a directions with equal velocities. Let 
mn be the front of a plane wave per- 
4 p____ pendicular to the plane of the paper, . 
m moving from left to right or towards 


Enis P. Draw PA perpendicular to the 
wave front, and draw Pa, Pd, etc., at such obliquities that Pa 
shall exceed PA by half a wave length, Pd exceed Pa by 
half a wave length, etc. We will designate the wave length 
by A. 

It is evident that the total effect at P will be the sum of the 
effects due to the small portions Aa, ad, etc. Since Fa is half a 
wave length greater than PA, and Pd half a wave length 
greater than fa, each point of ad is half a wave length farther 
from P than some point in Aa; hence elementary waves from 
ab will meet at Pwaves from Aa in the opposite phase. It 
appears, therefore, that the effects at Pof the portions ad and 
Aa are opposite in sign, and tend to annul each other. The 
same is true of dc and cd. But the effects of da and aé may 


274] PROPAGATION OF LIGHT. 399 


be considered as proportional to their lengths. Hence, by 
computing the lengths, we can determine the resultant effect 
atP. Let d4P=-x. From the construction, we have 


Ese MAR ee Ry 
oa a/ (@+5) Fas +S: 
Ab= Vatily—2 = V2xk+N 
Ac= Vet Rye = V3ek +R 
Mie Cer eat = 4rd 4h"; 
Sy mms Clee 


For light the values of A are between 0.00039 and 0.00076 
mm., and if + be taken as 1000 mm,, A’ will be very small in 
comparison to +A and may be omitted. The above formvlas 


then become, if “A be represented by /, 


Aa=lyVi; 
Ab=/V2; 
Aces 1V3; 
Ad=1V4; 
Cle = CUC,, 


and the several portions into which the wave front is divided 
are 


Vi fmt an WY 
Gait) OA LA ; 
DW aie We2y, 0,3 181; 
ed = K V4 — V3) = 0.2681. 


400 ELEMENTARY PHYSICS. [274 


Taking now the pairs of which the effects at P are opposite 
in sign, we find Aaa little more than twice ad, while dc and cd 
are nearly equal. It is evident also, that for portions beyond 
d, adjacent pairs will be still more nearly equal, and the effect 
at P, therefore, of each pair of segments beyond 6 almost van- 
ishes. The effect at Pis then almost wholly due to that por- 
tion:of Aa that is not neutralized by ad. But, taking the 
sreatest value of A, da = Vxi = Vo.76 = 0.87mm., a very 
small distance. Hence, under the conditions assumed, the 
effect at any point Pis due to that 
portion of the wave front near the 
foot of the perpendicular let fall 
from Pon the wave front. It may 
be demonstrated by experiment 
that the portions of the wave be- 
yond Aa neutralize each other. 
Suppose a screen zzz in the position 
shown in Fig. 102. The point P 
will be in shadow. If the darkness at P is due to interference 
as explained, light should be restored by suppressing the in- 
terfering waves. If a second screen be placed at m’n’ so as to 
cut off the waves proceeding from points above 8, waves from 
points between a and 6 will no longer be neutralized, and light 
should fall at A To test this conclusion the edge of a flat 
flame may be observed through a narrow slit in a screen. In- 
stead of the narrow edge of the flame, a broad luminous surface 
is seen, in which the brightness gradually diminishes from the 
centre towards the edges. Jf we consider the wave front just 
entering the slit, it will be seen that elementary waves proceed 
from all points of it, and the slit being very narrow it is only 
in very oblique directions that pairs of these waves can meet in 
opposite phases. Hence, light proceeds in oblique lines behind 
the screen, and from our habit of locating visible objects back 
along the lime of light entering the eye, the flame appears as a 


Fic. roe. 


275] PROPAGATION OF LIGHT. 401 


broad surface. It will be seen by referencé to Fig. 1o1 that 
the elementary wave that first reaches P is the 
one to which the disturbance there is principally 
due. Other waves arriving later find there the 
opposite phase of some wave that has preceded 
them. When the velocity in all directions is the 
same, the first wave to reach FP is the one that 
starts from the foot of a perpendicular let fall from 
P onthe wave front. Hence light is said to travel 
in straight lines perpendicular to the wave front. 
If, however, light does not move with equal 
velocities in all directions, the last statement is 
no longer true, as will be seen from Fig. 103. 
Here mn represents a wave front, proceeding 
towards Pin a medium in which the velocities in different 
directions are such that the elementary wave surfaces are ellip- 
soids. The ellipses in the figure may be taken as sections of 
these ellipsoids.. The wave first to reach P is not the one 
that starts from A at the foot of the perpendicular, but from 
A’. It is from A’ that P derives its light, and the line of 
propagation is no longer perpendicular to the wave front. 

It is important to note that the deductions of this section 
apply only where A is small in relation to x, so that A* may be 
neglected in comparison with 4A. With sound waves this is 
not true, and if a computation similar to that given above 
for light-waves be made for sound, not omitting 4’, it will 
be seen why there are no definite straight lines of sound and 
no sharp acoustic shadows. 

275. Principle of Least Time.—The above are only par- 
ticular cases of a law of very general application, that light in 
going from one point to another follows the path that requires 
least time. The reason is.that values in the vicinity of a mini- 
mum change slowly, and there will be a number of points in 


the neighborhood of that point from which the light-waves are 
26 


Fic. 103. 


402 BOEEMENITARY. Pi ¥ olGo: [276 


propagated to the given point in the least time, from which 
waves will proceed to that point in sensibly the same time, and, 
meeting in the same phase, combine to produce light. It is 
also true that values change slowly in the vicinity of a maxi- 
mum, and there are cases where the path followed by the light 
is determined by the fact that the time is a maximum instead 
of a minimum. 

276. Shadows.—An optical shadow is the space from which 
light is excluded by an opaque body. When the luminous 
source is a point, or very small, the boundary between the light 
and shadow is very sharp. When the luminous source is large, 
there is a portion of the space behind the opaque body, called 
the wmbra, which is in deep shadow, and surrounding this is a 
space which is in shadow with reference to one portion of the 
luminous source while it isin the light with reference to an- 
other portion. The space from which light is only partially ex- 
cluded is the penumbra. Fig. 104 shows the boundaries of 
the umbra and penumbra. It is evident that the light di- 


_minishes gradually from the outer boundary of the penumbra 
to the boundary of the umbra. 

277. Images by Small Apertures.—If light from a single 
luminous point pass through a small hole of any form, and fall 
on a screen at some distance, it produces a luminous spot of the 
same form as the opening. Light from several points will pro- 
duce several such spots. If the luminous source bea surface, 
the spots produced by the light from its several points will 


277] PROPAGATION OF LIGHT. 403 


overlap each other and form an illuminated surface, which, if 
the source be large in comparison with the opening, will have 
the general form of the source, and will be inverted. The illu- 
minated surface is an inverted zmage of the source. If a small 
opening be made in the window-shutter of a darkened room, 
images of external objects will be seen on the wall opposite. 
The smaller the opening, the more sharply defined, but the less 
brilliant, is the image. 


CHAP E Ratt 
REFLECTION AND REFRACTION. 


278. Law of Reflection.—In § 246, it is shown that when 
a wave passes from one medium into another where the parti- 
cles constituting the wave move with greater or less facility, a 
wave is propagated back into the first medium. It is shown in 
§ 247, that when the surface separating the two media is a plane 
surface, the centres of the incident and reflected waves are on 
ps the same perpendicular to the sur- 
face, and at equal distances on oppo- 
site sides. Considering the lines to 
which, as shown in § 274, the wave 
propagation in the case of light is re- 
stricted, a very simple law follows. 
at once from this relation of the 
incident and reflected waves. In 
Fig. 105, C and C’ represent the centres of the incident and re- 
flected waves mn,on. CA, AB are the paths of the incident 
and reflected light. It will be evident from the figure that 
CA, AB are in the same plane normal to the reflecting surface, 
and that they make equal angles with the normal AV. CAN 
is called the angle of incidence, and VAZS the angle of reflec- 
tion. Hence we may state the Jaw of reflection as follows: 
The angles of incidence and reflection are equal, and lie in the 
same plane normal to the reflecting surface. It can easily be 
shown that light traverses the path CAB from C to B which 
fulfils these laws, in less time than it requires to traverse any 
other path by way of the reflecting surface. 


Cc 

| 

| 

| 

I 

| 

| 
c’ 


Fic. 105. 


279] REFLECTION AND REFRACTION. 405 


279. Law of Refraction.—lf the incident wave pass from 
the one medium into the other, there is in general a change in 
the wave front, and a consequent change in the direction of the 
light. Let us first consider the simple case of a plane wave en- 
tering a homogeneous, isotropic medium of which the bounding 
surface is plane. Suppose both planes perpendicular to the 


Fic. 106. 


plane of the paper, and let AB (Fig. 106) represent the 
intersection of the surface of the medium, and mz the in- 
-tersection of the wave, with that plane. Let vw represent the 
velocity of light in the medium above AB, and ~ the ve- 
locity in the medium below it. Let m’o be the position 
of the wave in the first medium after a time ¢. Then mo 
equals vf. As the wave front passes from mz to m’o, the 
points of the separating surface between z and o are succes- 
sively disturbed, and become centres of spherical waves propa- 
gated into the second medium with the velocity v’. The wave 
surface of which the centre is # would, at the end of time, 
/ 
have a radius zz’’=v’'¢, such that i or = Similarly, the 
un Vv 
wave from any other point, as s, would have a radius sz’ such 


a7 v ee 
that aioe and the wave surface within the second medium 


406 ELEMENTARY PHYSICS. [275 


is evidently the plane oz’’. As the direction of propagation 
is perpendicular to the wave front, of will represent the direc- 
tion of the light in the second medium. In the triangles zoz’ 
and zon’ we have zn’ = no sin Aon’, and nn” = no sin Aon’; 
hence 
sin Aon’ NH Ne 
sin Aon” — nn!’ v” 
If we represent the angle of incidence moN by z, and the 
angle of refraction poN’ by r, we have 
Soe supe 
any gl mM @ constant. (109) 


This constant is called the zudex of refraction. This is the 
expression of Snell’s daw of refraction. Here again the time 
required for the light to pass by mop from m in one medium 
to f in the other is less than by any other path. 

We may now trace a wave through a medium bounded by 
plane surfaces. Suppose the wave front and bounding planes. 
of the medium all perpendicular to the plane of the paper. 


sin, 2h ae 
We shall have as above for the first surface —— =e 
sine iee 
. 4 i! 
Sin vv 
and /for the second surlace 5 
Sin 7h py 


If, as is often the case, the light emerge into the first me- 
dium, 


v 
iim Kata DE cay a re (110) 


If the bounding planes be parallel, z’ = 7, and we have 


sin 7 


I 
STL ee 


407 


hence z= 7’, or the incident and emergent waves are parallel 
If the two bounding planes form an angle A the body is 
called a prasm. The wave incident upon the second face will 
make with it an angle A —~7, 
and the emergent waveis found 
by the relation 


279] REFLECTION AND REFRACTION. 


hci a Ame I sin 7” 


SHR) fe Se .sin(A—r) *" 


ni 


x 
\ 
\ 
\ 
The direction of the emerging \ 
wave front may be found by \ 
| \ 

\ 

\ 


r \ 
2 hi Hh Vi J 
construction. fe VA 

Draw Az (Fig. 107) parallel BN Ce a 
to the incident wave. From \ i 

: . \ rt 

some point 4 on AB describe x a 
an arc tangent to Az; from the asec 


; Merwe) 
same point with a radius i describe the arc vv. Az, tangent 


to rr, is the refracted wave front. From some point C on AC 
describe an arc tangent to Az, and from the same point as cen- 
tre describe another arc 7’7’ with a radius ux Cy. A tan- 
gent from A to 77’ is parallel to the emergent wave. It might 
be that A would fall inside the arc 7’7’ so that no tangent 
could be drawn. That would mean that there could be no 
emergent wave. The angle of incidence for which this occurs 
can readily be obtained from Eq. (110). We have 


I . / e hd 

at Ole Sill: ke ee Sil 20. 

fh 

Now the maximum value of sin” is 1, which is reached when 


sine’ =-—. Any larger value of sinz’ gives an impossible value 
Va ’ 


408 ELEMENTARY PHYSICS. [279 


} : ae +s 
forsin'y ob heangie Zu sine ye is called the crztical angle of 


the substance. For larger angles of incidence the light cannot 
emerge, but is ¢otally reflected within the 
medium. 

Another construction for the front of 
the emergent wave is very instructive. 
Let AS, AC (Fig. 108), be theviageuuan 
the prism, and let Az drawn through A 
be parallel to the front of the incident 
wave. With JA as centre, and any radius, 

\ draw an arc 7m. From the same centre 


—— 
Sa 


At 
Fic. 108. with. radius Are na describe another 


arc. From 7 draw rx parallel to AB and join Ax. A* is 
parallel to the front of the refracted wave. For in the triangle 
Arz we have ; 


sin Arx SInie7y ae 4 
Per G7 gis — Ape Pe = ye = J, by construction. 

Since zrx equals the angle of incidence, Axr equals the 
angle of refraction. Now draw zr’ parallel to AC, and Ar’ 
is parallel to the front of the emergent wave. The angle Av 
is the deviation that the wave suffers in passing through the 
prism. Suppose the prism to rotate about A and the angle of 
incidence to change in such a way that the condition of 
things may be always represented by rotating the angle 717’, of 
which the sides are parallel to the sides of the prism, around x. 
It is plain that the arc 7’ will be longer or shorter as it crosses 
the angle more or less obliquely, and that its length will be a 
minimum when 27’ and #r are equal—that is, when the line Ax 
bisects the angle at x and consequently the angle 4 of the 
prism. But the arc 7’r may be taken as the measure of the 


280] REFLECTION AND REFRACTION. 409 


angle of deviation 7’Ar at its centre. Hence that angle is a 
minimum when it is bisected by Ax, and when, therefore, the 
angles of incidence and of emergence are equal. Considering 
that the path of the light is perpendicular to the wave front, 
the above construction shows that the deviation, when yu is 
greater than unity, is always toward the thicker portion of the 
prism. The case when emergence is no longer possible is also 
shown by the failure of x7’, parallel to AC, to cut the arc 7’z. 
The critical angle is reached when +7’ becomes tangent to7’s. 
If, in a prism of any substance, +7 and +r’ be both tangent to 
rr, the angle of that prism is the greatest angle which will ad- 
mit of the passage of light through the prism. 

If a beam of white light be allowed to fall upon a prism 
through a narrow slit, it will be refracted, in general, in accord- 
ance with the law already given. The image of the slit, how- 
ever, when projected upon a screen, appears not as a single line 
of white light, but as a variously colored band. This is due to 
the fact that the indices of refraction for light of different 
colors are different. Hence the index of refraction of a sub- 
stance, as ordinarily given, depends upon the color of the light 
used in determining it,and has no definite meaning unless that 
color is stated. 

280. Plane Mirrors.—The wave on, represented in Fig. 
105, is the same as would have come from a luminous point at 
C’ if the reflecting surface did not intervene. If this wave 
reach the eye of an observer, it has the same effect as though 
coming from such a point, and the observer apparently sees a 
luminous point at C’. C’ isa virtual tmage of C. When an 
object is in front of a plane mirror each of its points has an 
image symmetrically situated in relation to the mirror, and 
these constitute an image of the object like the latter in all 
respects, except that by reason of symmetry it is reversed in 
one direction. 

The reflected light may for all purposes be considered as 


410 ELEMENTARY PHYSICS. [28r 


coming from the image. If it fall on a second mirror and be 
again reflected, a second image appears behind this mirror, the 
position of which is determined by considering the first image 
as an object. When two mirrors make an angle, an object 
between them will have a series of images, as shown in Fig. 
109. AB#and AC represent the intersections of the two mir- 
rors with the plane of the paper, to which they are supposed 
perpendicular. O is the object. It will 
have an image produced by AJB, the 
position of which is found by drawing 
e OO perpendicular to AB and making 
,mb= m0. The light reflected from 
1° AB proceeds as though 6 were the ob- 
ject, and falling on AC is again reflect- 
ee ee ed, giving an image at ¢’. Proceeding 
bc” from AC, it may suffer a third reflection 
weak from AZ and give a third image at 6”. 
With the angle as in the figure none of the light can suffer a 
fourth reflection, because after. the third reflection the light 
proceeds as though originating at 6’, and 6” is behind the 
plane of the mirror dC. Images ¢, 0’,and c”’ are produced by 
light which suffers its first reflection from AC. It is easy to 


° C 


show that all these points are equidistant from A, and hence 


are on the circumference of a circle of which J is the centre. 
If OAC were an even aliquot part of four right angles, c’’ and 
6” would coincide, and the whole number of images, including 
the object, would be the quotient of four right angles by the 
angle formed by the mirrors. This is the principle of the 
kaleidoscope. 

281. Spherical Mirrors.—A spherical mirror is a portion 
of a spherical surface. It is a concave mirror if reflection 
occur on the concave or inner surface; a convex mirror if it 
occur on the convex surface. The centre of the sphere of 
which the mirror forms a part is its centre of curvature. The 


281] REFLECTION AND REFRACTION. 41! 


middle point of the surface of the mirror is the vertex. A line 
through the centre of curvature and the vertex is the przuczpal 
axis. Any other line through the centre of curvature is a 
secondary axis. The angle between radii drawn to the edge 
of the mirror on opposite sides of the vertex is the aperture. 
To investigate the effects of reflection from a spherical surface, 
let us consider first a concave mirror. Let a light-wave ema- 
nate from a point Z on the principal axis (Fig. 110). In general, 


Fic, 110, 


different points of the wave will reach the mirror successively, 
and, considering the elementary waves that proceed in turn 
from its several points, the reflected wave surface may be con- 
structed as fora plane mirror. If the mirror were not there 
the wave front would, at a certain time, occupy the position 
aa. Drawing the elementary wave surfaces we have 00, the 
position at that instant of the reflected wave. Its form sug- 
gests that of a spherical surface, concave toward the front, and 
having a centre at some point Z on the axis. If we assume it 
to be so, and try to determine by analysis the position of 4a 
real definite result will be proof of the correctness of our 
assumption. If 44 be a spherical surface and @ its centre, it is 


412 ELEMENTARY PHYSICS. [281 


plain that the disturbances propagated from the various points 
of 6@ will reach 7 at the same instant, and / will at that instant 
be the wave front. It is plain, too, that the time occupied by 
the wave in going from the radiant point to all points of the 
same wave front must be the same. Hence, in a homogeneous 
medium, the length of path to the various points of the wave 
must be constant, that is, in the case under consideration, 
LB -4-£&b must be constant for all points of the wave front 4d. 
If Z be a subsequent position of 04, it follows that LB+ 47 
must be constant wherever the point & is situated on the re- 
flecting surface. Draw 4D perpendicular to the axis of the 


mirror. Represent BD by y, AD by «, LA by peyatiaee 


and CA by rv Then we have LB=V(p—-+)’+y7’, and 
yf = (2r — x)x = 2rx — x’. Hence follows 


LB=Vp —2pr+xtorxr—x 
= Vp" + 24(r — p). 


If the aperture be small, 2 will be small in comparison 
with the other quantities, and we may obtain the value of LP 
to a near approximation by extracting the root of the ex- 
pression found above and omitting terms containing the second 
and higher powers of «. We obtain 


LB=p+F(r—p)+.... 


In like manner we have 
/ + / 
No erm anya aia ile rier 


whence LB+/B=pt+p’'+ AG — p) ag — ify 


7 


281] REFLECTION AND REFRACTION. 413 


When & coincides with A, the above value becomes p + J’, 
and since upon our supposition all values of L&+ /B are 
equal, we must have 


p+P =Pt 0 4500- A450?) 


from which we obtain 


niin, 
5 ieee 
and Bee 


As this is a definite value, it follows that, for the apertures 
for which the approximations by which the result was arrived 
at are admissible, the wave surface is practically spherical. Since 
the disturbances propagated from 60 reach / simultaneously, 
their effects are added, and the disturbance at Z is far greater 
than at any other point. The effect of the wave motion is 
concentrated at / and this point is therefore called a focus. 
Since the light passes through 4 it is a veal focus. If 2 were 
the radiant point, it is clear that the reflected light would be 
concentrated at Z.. These two points are therefore called con- 


jugate foct. If we divide both sides of the equation 5+ 5 i, 
by 7, we have 

Te GLa les.2 

Py et Be? 


which is the usual form of the. equation used to express the 
relation between the distances from the mirror of the conju- 
gate foci. 


“414 ELEMENTARY PHYSICS. [28r 


A discussion of this equation leads to some interesting 
results. Suppose p= oo, then p’ = 47;/ that 1s, whenethe 
radiant is at an infinite distance from the mirror, the focus is 
midway between the mirror and the centre. In this case the 
incident wave is normal to the principal axis, and the focus is 
called the principal focus. Suppose p=7; p=~r also. When 
Pics ah, VP sot. Wl WHE pins ~, 5 > >and g=5-5F a 
negative quantity. To interpret this negative result it should 
be remembered that all the distances in the formulas were 
assumed positive when measured from the mirror toward the 


Sey 
act 
~ 


ee St 
~. 
~~ 


DiGurrr. 


source of light. A negative result means that the distance 
must be measured in the opposite direction, or behind the 
mirror. Fig. 111 represents this case. It is evident that 
the reflected wave is convex toward the region it is ap- 
proaching, and proceeds as though it had come from JZ. 
Z is therefore a virtual focus. Either of the other quantities 
of the formula may have negative values. jp will be negative 
if waves approaching their centre / fall on the mirror. Plainly 
they would be reflected to Z at a distance from the mirror less 


; 
than 57 as may be seen from the formula. If 7 be negative, 


the centre is behind the mirror. The mirror is then convex, 
and the formula shows that for all positive values of J, Jf’ is 
negative and numerically smaller than #. 


282] REFLECTION AND REFRACTION. 415 


282. Refraction at Spherical Surfaces.—The method of 
discussion which has been applied to reflection may be em- 
ployed to study refraction at spherical surfaces. Let BD 
(Fig. 112) be a spherical surface separating two transparent 


Fic. 112. 


media. Let wv represent the velocity of light in the first 
medium, to the left, and v’ the velocity in the second medium, 
to the right, of BD. Let Z be a radiant point, and mz a sur- 
face representing the position which the wave surface would 
have occupied at a given instant had there been no change in 
the medium, 7z’z’ the wave surface as it exists at the same 
instant in the second medium in consequence of the different 
velocity of light in it. Assume as before, in § 281, that mn’ 
is a spherical surface with centre 7. We have 


sid Jad 
Vv ey rahe 
aoe Dea em ee 
Vv Vv Vv 


a constant for all points of mx. If 7 be the centre of the 
spherical surface e’n', we have 


A16 ELEMENTARY PHYSICS. [282 


‘B . Bu’ 
TaN + jini Cis 


“Ue Vv 


a constant for all points of #’x’. | 
Taking the difference of the last two equations, and re- 
membering that 


Lihasd Brae 
v yy! , 
we obtain seal ee me (OME G4 
v v 


a constant for all points of BD, and hence 
LL — “1B =a constant. 


v ; ; , 
But Bu parign nite the index of refraction of the second sub- 


stance in relation to the first. Hence ZB — p/B =a constant 
= LA— wlA. Using the notation of the last section, and 
substituting the values of Z& and /B&as there found, except 
that p” is used instead of f’, we have 


e+ ir-p)- ip" +r 2) =p — 18" 


rv r 
whence we obtain 2 ae =I—4, 
I ee 
and ra igs “ : i (112) 


282] REFLECTION AND REFRACTION. 417 


If the medium to the right of BD be bounded by a second 
spherical surface, it constitutes a /ezs. Suppose this second 
surface to be concave toward / and to have its centre on AC. 
The wave mn’, in passing out at this second surface, suffers a 
new change of form precisely analogous to that occurring at 
the first surface, and the new centre is given by the formula 
just deduced by substituting for # the distance of, the wave 
centre from the new surface, and for yu the index of refraction 
of the third medium in relation to the second. If s represent 
the distance of / from the new surface, w’ the new index, and 
p’ the new focal distance, we have 


/ 


I w—t 
p’ Saas a 20h te 


If we suppose the lens to be very thin we may put s= pg”. 
If we suppose also that the medium to the right is the same 


; I 
as that to the left of the lens, yw’ is equal to i Hence 


I I 
— ——I 
Ae seal had 
p p’ x! is 
Multiplying through by y, we have 
Ly (sly a ee 
De AO ae i arr ae y! 


Eliminating pp” between this equation and Eq. 112, we obtain 


I I I I 
ped ste AU am (> —5) (113) 
27 


418 ELEMENTARY PHYSICS, [282 


which expresses the relation between the conjugate foci of the 
lens. It should be noted that 7 in the above formulas rep- 
resents the radius of the surface on which the light is incident, 
and 7’ that of the surface from which the light emerges. All 
the quantities are positive when measured toward the source 
of light. Fig. 113 shows sections of the different forms of 


al 2 3 4 5 6 
Fic. 113. 


lenses produced by combinations of two spherical surfaces, or 
of one plane and one spherical surface. 

. An application of Eq. 113 will show that for the first three, 
which are thickest at the centre, light is concentrated, and for 
the second three diffused. The first three are therefore called 
converging, and the second three diverging, lenses. Let us 
consider the first and fourth forms as typical of the two classes. 
The first is a double convex lens. The 7 of Eq. 113 is nega- 
tive because measured from the lens away from the source of 
light. The second term of the formula has therefore a negative 

' “eel I I 
value, and #’ is negative except when — > (u— y(t o a 


P 


I I I : 
—= = ("—1) oi ee negative 


P 


quantity because 7 is negative. /’ is then the distance of the 
principal focus from the lens, and is called the focal length of 
the lens. The focal length is usually designated by the sym- 
bol f. Its negative value shows that the principal focus is on 
the side of the lens opposite the source of light. This focus 
is real, because the light passes through it. Eq. 113 is a little 
more simple in application if, instead of making the algebraic 


If p = 2, we have = =o and 


283] REFLECTION AND REFRACTION. 419 


signs of the quantities depend on the direction of measure- 
ment, they are made to depend on the form of the surfaces 
and the character of the foci. If we assume that radii are 
positive when the surfaces are convex, and that focal distances 
are positive when foci are real, the signs of f’ and v in Eq. 113 
must be changed, since in the investigation 7’ is the distance 
of a virtual focus, and 7 the radius of a concave surface. The 
formula then becomes 


erat hee 

St+5=— (7 +5). (114) 
To apply this formula to a double concave lens, 7 and 7’ 

are both negative; 7’ is then negative for all positive values of 

p. That is, concave lenses have only virtual foci. For a 

plano-convex lens (Fig. 113, 2), if light be incident on the 

plane surface, 


I 1H) OX 
fia poy and oe tee 


es 
This gives positive values of f’ and real foci for all values of 


I I 
fran Alera 8 ie, 

For a concavo-convex lens (Fig. 113, 6) the second member 
of the equation will be negative, since the radius of the con- 
cave surface is negative and less numerically than that of the 
convex surface. Hence jf’ is always negative and the focus 
virtual when Z is real. | 

283. Images formed by Mirrors.—In Fig. 114 let ad rep- 
resent an object in front of the concave mirror zzz. We know 
from what precedes that if we consider only the light incident 


420 ELEMENTARY: PITY SLCS, [283 


not too far from c, the light reflected will be concentrated 
at some point a’ on the axis ac at a distance from the mirror 
given by Eq. 114. a@’ isa real 


6’ is an image of J. If axes 
were drawn through other 
points of the object, the im- 
ages of those points would be 
found in the same way. They 

Fic. 114. would lie between a’ and 0’, 
and a’d’ is therefore a real image of the object. It is inverted,. 
and lies between the axes ac, dd, drawn through the extreme 
points of the object. ‘The ratio of its size to that of the ob- 
ject is seen from the similar triangles adC, a’b’C, to be the 
ratio of the distances from C. From Eg. 111 we obtain 


p' 7.) PONT erne 


p” pr pr 


Since 7 — f’ and p —, are respectively the distances from 
the centre of the image and object, we have 


or, the image and object are to each other in the ratio of their 
respective distances from the mirror. As the object approaches, 
the image recedes from the mirror and increases in,size. At 
the centre of curvature the image and object are equal, and 
when the object is within the centre and beyond the principal 
focus the image is outside the centre and larger than the ob- 
ject. When the object is between the principal focus and the 
mirror, the image is virtual and larger than the object. Con- 
vex mirrors produce only virtual images, which are erect and 
smaller than the object. 


zmage of a. In the same way 


285] REFLECTION AND REFRACTION. 421 


284. Images formed by Lenses.—Let us suppose an ob- 
ject in front of a double convex lens, which may be taken as 
a type of the converging lenses. The point ¢ (Fig. 115) will 
have an image at the conjugate : 
focus on the principal axis. @ ‘ ot 
and 6 will have images on 
secondary axes drawn through 
those points respectively, and 4% 
a point called the optical cen- 
tre of the lens. So long as these secondary axes make but a 
small angle with the principal axis, definite foci will be formed 
at the same distances as on the principal axis, and an image 
ab’ will be formed which will be real and inverted, or virtual 
and erect, according to the distance of the object from the lens. 
The formula 


al 


Bigs. 


I I 


stza-n(+s)=5 


eld g 

shows that when # increases f’ diminishes, and conversely. It 
shows also that when 7 is less than 7, 7’ is negative, and the 
image virtual. It is plain from the figure that the sizes of image 
and object are in the ratio of their distances from the lens. 
Diverging lenses, like diverging mirrors, produce only virtual 
images smaller than the object. 

285. Optical Centre.—It was stated in the last section that 
the secondary axes of a lens pass through a point called the 
optical centre. The location of this point is determined as fol- 
lows ely Figs HOw let Cie ibe 
the centres of curvature of the 
two surfaces of the lens, and let 
CA and C’B be two parallel 

Fic, 116. radii,, \ [hes tangents at “As and 
B are also parallel, and light entering at B and emerging at 4 
is light passing through a medium with parallel surfaces (§ 279), 


422 ELEMENTARY PHYSICS, [286 


and suffers no deviation. If wedraw AB, cutting the axis at O, 


the triangles CAO, CBO are similar, and c= os But a 


being the ratio of the radii, is constant for all parts of the sur~ 


Cy 
faces, hence == must be constant, or all lines such as AB must 


CO 
cut the axis at one point O. O is the optical centre, and light 
passing through it is not deviated by the lens. 

286. Geometrical Construction of Images.—For the 
geometrical construction of images formed by curved surfaces, it 
is convenient to use, in place of the waves themselves, lines per-. 
pendicular to the wave front, which represent the paths which 
the light follows, and are called vays of light. ‘These rays, 
when perpendicular to a plane wave surface, are parallel, and an 
assemblage of such rays, limited by an aperture in a screen, is 
called a beam. When the rays are perpendicular to a spherical 
wave surface, they pass through the wave centre, and constitute 
a pencil. 

A plane wave surface perpendicular to the axis of a lens 
is converted by the lens into a spherical wave surface with its 
centre at the principal focus. The rays perpendicular to the 
plane wave surface are parallel to the axis, and after emergence 
must all pass through the principal focus. Conversely, rays 
emanating from the principal focus emerge from the lens as 
| rays parallel to the axis. Also, 
rays emanating from any focus. 
must, after emerging from the 
lens, meet at the ‘conjugate 
focus. Let,2, Kigveiige 
converging lens, and AB an 
object. Let O be the optical centre, and / the principal focus. 
Since all the rays from A must meet, after emerging from the 
lens, at the conjugate focus, which is the image of A, to find the 
position of the image it is only necessary to draw two such rays 


288] REFLECTION AND REFRACTION. 423 


and find their intersection. The ray through the optical centre 
is not deviated, and the straight line 4A’ represents both the in- 
cident and emergent rays. The ray 4 Z may be considered as one 
of a group parallel to the axis. All such rays must, after passing 
through the lens, pass through the principal focus. LA’, passing 
through Ff, is therefore the emerging ray, and its intersection 
with AA’ locates the image of A. Hence, to construct the 
image of a point, draw from the point two incident rays, and 
determine the corresponding 
emergent rays. The intersec- 
tion of these will determine the 
image. The rays most conve- 


the optical centre and the ray 
parallel to the axis or through the principal focus. Fig. 118 
gives another example of an image determined by construction. 

287. Thick Lenses.—When a lens is of considerable thick- 
ness, the formula derived in § 282 does not give the true posi- 
tion of the conjugate foci. A formula involving the thickness 
of the lens may be derived without difficulty, but for practical 
purposes it is usual to refer all measurements to two planes, 
called the principal planes of the lens. The determination of 
the position of these planes involves a discussion which does 
not come within the scope of this book. 

288. Mirrors and Lenses of Large Aperture.—The 
equations derived in §§ 281, 282, are only approximations, ap- 
plying with sufficient exactness to mirrors and lenses of small 
aperture. But for large apertures, terms containing the higher 
powers of x cannot be neglected, + will not disappear from the 
expression of yp’, and p’ will, therefore, not have a definite 
value. In other words, the reflected or refracted wave is not 
spherical, and there is no one point / where the light will be 
concentrated. Surfaces may, however, be constructed which 
will, in certain particular cases, produce by reflection or refrac- 


A24 ELEMENTARY PAYSICS. [288 


tion perfectly spherical waves. If we desire to find a surface 
ie such that light from Z (Fig. 119) is con- 
centrated by reflection at /, we remem- 

ifm ber that the sum LA4-+ 4S/ must be 

L constant, and that this is a property of 

On: an ellipse with foci at Zand Z. If the 
ellipse be constructed and revolved about L/ as an axis, it will 
generate a surface which will have the required property. If 
one of the points Z be removed to an infinite distance, the 
corresponding wave becomes a plane perpendicular to L/, and 
we must have L&-+ LC (Fig. 120) constant, 
a property of the parabola. A _ parabolic 
mirror will therefore concentrate’ at its focus 
incident light moving in paths parallel to its 
axis, or will reflect incident light diverging 
from its focus in plane waves perpendicular 
to its axis. 

Mirrors and lenses having surfaces which Free 450, 
are not spherical are seldom made because of mechanical diffi- 
culties of construction. It becomes necessary, therefore, to 
consider how the disadvantages arising from the use of spheri- 
cal surfaces of large aperture for reflecting or refracting light 
may be avoided or reduced. 

We will consider first the case of aspherical mirror. It was 
shown above that light from one focus of an ellipsoid is reflected 
from the ellipsoidal surface in perfectly spherical waves concen- 
tric with the other focus. Let Fig. 121 represent a plane sec- 
tion through the axis of an ellipsoid, and /ca a small incident 
pencil of light proceeding from the focus /% Fac is a section 
of the reflected pencil. It is a property of the ellipse that the 
normals to the curve bisect the angles formed by lines to the 
two foci. The normal ae bisects the angle /aF’, and hence in 
Laduure 
Fa’ Fe 


the triangle Fal’ we have 


288 | REPLECLTION AND “REFRACTION, 425 


If a’ move toward ¢, Fa increases and Fa diminishes. Hence, 
from the above proportion, /’e must increase and Fe diminish ; 
or, the successive normals as we approach the minor axis cut 
the major axis in points successively nearer the centre of the 
ellipse. The normals produced will therefore meet each other 
at z beyond the axis. If ac be taken small enough it may be 
considered the arc of acircle of which az, cn are radii and x 
the centre. It is therefore a meridian section of an element 
of a spherical surface of which /7 is an axis. 

Sections of wave surfaces reflected from the ellipsoid have 
their centre at #4”, and are also sections of wave surfaces re- 
flected from the elementary spherical surface. Evidently the 
same would be true for any other meridian section passing 


Pic. 12%: 


through FA of the sphere of which the elementary surface 
forms a part, and the form of the wave surfaces may be con- 
ceived by supposing the whole figure to revolve about /A as 
an axis. The arc ac describes a zone of the sphere, s,s, 7, 7, 
describe wave surfaces, and /” describes a circumference having 
its centre on #A. The wave surfaces are portions of the sur- 
faces of curved tubes of which the axis is the arc described by 
the point F’. The line described by /” is a focal fine, and all 
the light from the zone described by ac passes through it, or 
does so very approximately. If ac be taken nearer to 4 on the 
sphere, F” approaches the axis along the curve #’/” and finally 


426 ELEMENTARY PHYSICS. [288 


coincides with /”’, the focus conjugate to /& F’F” is a caustic 
curve, which, when the figure revolves about the axis AF, 
describes a caustic surface. It will be noted that all the light 
from the zone described by ac passes through the axis AF be- 
tween the points «and y. The light coming from / and re- 
flected from a small portion of the spherical surface around 8, 
the middle point of ac, is then concentrated first in a line 
through /’ at right angles to the paper, and again into the line 
ay in the plane of the paper. Nowhere is it concentrated into 
a point. A line drawn through 6 and the middle of the focal 
line through /” is the axis of the reflected pencil. It will in- 
tersect the axis of the mirror between x and y. If a plane be 
passed through the point of intersection perpendicular to the 
axis of the pencil, its intersection with the pencil will be like 
an elongated figure 8, which may be considered as a focal line | 
at right angles to the axis of the pencil, and in the plane of the 
paper, and therefore at right angles to the focal line through 
F’, Between these two focal lines there is a section of least 
area, nearly circular, which is the nearest approach to an image 
of / produced by an oblique incidence such as we have been 
considering. , 

If refraction instead of reflection had taken place at ac,a 
result very similar would have been obtained for the refracted 
pencil. This failure of spherical reflecting or refracting sur- 
faces to bring the light exactly to a focus is called spherical 
aberration. In order to obtain a sharp focus, therefore, if only 
a single spherical surface be employed, the light must be con-— 
fined within narrow limits of normal incidence. When reflec- 
tion or refraction takes place at two or more surfaces in succes- 
sion, the aberration of one may be made to partially correct 
the aberration of the other. For instance, when the waves in- 
cident upon a double convex lens are plane, the emerging 
waves are most nearly spherical when the radius of the second 
surface is six times that of the first. [wo or more lenses may 


290] REFLECTION AND REFRACTION. 427 


be so constructed and combined as to give, for sources of light 
at a certain distance, almost perfectly spherical emerging 
waves. Such combinations are called apl/anatic. The same 
term is applied to single surfaces so formed as to give by re- 
flection or refraction truly spherical waves. 


SIMPLE OPTICAL INSTRUMENTS. 


289. The Camera Obscura. — If a converging lens be 
placed in an opening in the window-shutter of a darkened 
room, well-defined images of external objects will be formed 
upon a screen placed ata suitable distance. This constitutes 
a camera obscura. The photographer’s camera is a box in one 
side of which is a lens so adjusted as to form an image of ex- 
ternal objects on a plate on the opposite side. The relation 
deduced in $284 serves to determine the size of the image 
which a given lens will produce, or the focal length of a lens 
necessary to produce an image of a cettain size. 

290. The Eye as an Optical Instrument.— The eye, as 
may be seen from Fig. 122, which represents a section bya 
horizontal plane, is a camera obscura. a@ is a 
transparent membrane called the cornea, be- 
hind which is a watery fluid called the aqueous 
humor, filling the space between the cornea 
and the crystalline lens. Behind this is the 
vitreous humor, filling the entire posterior 
cavity of the eye. The aqueous humor, crys- 
talline lens, and vitreous humor constitute a a 
system of lenses, equivalent to a single lens of AES 
about two and a half centimetres focus, which produces a real 
inverted image of external objects upon a screen of nervous 
tissue called the retina, which lines the inner surface of the 
posterior half of the eyeball. The retina is an expansion of the 
optic nerve. The light that forms the image upon it excites the 


428 ELEMENTARY PHYSICS. [290 


ends of the nerve, and, through the nerve-fibres leading to the 
brain, produces a mental impression, which, partly by the aid 
of the other senses, we have learned to interpret as the charac- 
teristics of the object the image of which produces the impres- 
sion. For distinct vision the image must be sharply formed on 
the retina; but as an object approaches, its image recedes from 
a lens, and if, in the eye, there were no compensation, we could 
see distinctly objects only at one distance. The eye, however, 
adjusts itself to the varying distances of the object by chang- 
ing the curvature of the front surface of the crystalline lens. 
There is a limit to this adjustment. For most eyes, an object 
nearer than fifteen centimetres does not have a distinct image 
on the retina. 

We may here consider the means by which we estimate the 
distance and size of an object. The retina is not all equally 
sensitive. The depression at 0, called the yellow spor, is much 
more sensitive than the other portions, and a minute area in 
the centre of that depression is much more sensitive than the 
rest of the yellow spot. That part of an image which falls on 
this small area is much more distinct than the other parts. 
How small this most sensitive area is, can be judged by care- 
fully analyzing the effort to see distinctly the minute details of 
an object. For instance, in looking at the dot of an za 
change can be detected in the effort of the muscles that con- 
trol the eyeball, when the attention is directed from the upper 
to the lower edge of the dot. The eye can then be directed 
with great precision to a very small object. The line joining 
the centre of the crystalline lens with the centre of the sensi- 
tive spot may be called the optic axis; and when the attention 
is directed to any particular point of an object, the eyeballs 
are turned by a muscular effort, until both the optic axes pro- 
duced outward meet at the point. For objects at a moderate 
distance we have learned to associate a particular muscular 
effort with a particular distance, and our judgment of such 


291] REFLECTION AND REFRACTION. 429 


distances depends mainly on this association. The angle be- 
tween the optic axes when they meet at a point is called the 
optic angle. Our estimate of the size of an object is based on 
our judgment of its distance, together with the angle which 
the object subtends at the eye, called the wzswal angle. In Fig. 
123, when ad is an object, and / the crystalline lens, @ is the 
visual angle. It is plain that the size of the image on the 
retina is proportional to the visual angle. It is plain, too, that 


/)’ 
W 


Fic. 123. 


an object of twice the size, at twice the distance, would sub- 
tend the same visual angle and have an image of the same 
size as ab. Nevertheless, if we estimate its distance correctly 
we shall estimate its size as twice that of ad; but if in any way 
we are deceived as to its distance, and judge it to be less than 
it really is, we underestimate its size. Most persons underes- 
timate heights, and hence underestimate the sizes of objects 
high above them. The visual angle is the apparent szze of the 
object. 

291. Magnifying Power.—To increase the apparent size 
of an object, and so improve our perception of its details, we 
must increase the visual angle. This can be done by bringing 
the object nearer the eye, but it is not always convenient or 
possible to bring an object near, and even with objects at hand 
there is a limit to the near approach, due to our inability to 
see distinctly very near objects. Certain optical instruments 
serve to increase the visual angle, and so improve our vision. 
Instruments for examining small objects, and increasing the 


430 ELEMENTARY PHYSICS. ~ [292 


visual angle beyond that which the object subtends at the 
nearest point of distinct vision by the unaided eye, are called 
microscopes. Those used for observing a distant object and 
enlarging the visual angle under which it is seen at that dis- 
tance are felescopes. In both cases the ratio of the visual angles, 
as the object is seen with the instrument, and without it, is the 
magnifying power. 

292. The Magnifying Glass.—Fig. 124 shows how a con- 

» verging lens may be employed 
to magnify small objects. The 
point a of an object just inside 

_ the principal focus ¥ of the lens 
“ A is the origin of light-waves 
6” which, after passing through the 
lens, are changed to waves hav- 
ing a centre a’ (§ 282) which, 
; when the lens is properly ad- 

Fic. 124. justed, is at the distance of dis- 
tinct vision. Waves coming from 6 enter the eye as though 
from 6’. The object is therefore distinctly seen, but under a 
visual angle a’Od’, while, to be seen distinctly by the unaided 
eye, it must be at the distance 
Oa’, when the angle subtended 
is a’’O0b"". The ratio of these an- 
gles is very nearly that of Oa’ 
to OF. Hence the magnifying 
power isthe ratio of the distance 
of distinct vision to the focal 
length of the lens. 

293. The Compound Mi- 
croscope.—A still greater mag- 
nifying power may be obtained 
by first forming a real enlarged image of the object (§ 284) and 
using the magnifying glass upon the image, as shown in Fig. 125. 


Fic. 125. 


294] REFLECTION AND REFRACTION. 431 


The lens A is called the objective, and E£ is called the eyve-lens or 
ocular. As will be seen in § 310, both 4 and £& often consist of 
combinations of lenses for the purpose of correcting aberration. 

294. Telescopes.—lIf a lens or mirror be arranged to pro- 
duce a real image of a distant object, either on a screen or in 
the air, we may observe the image at the distance of distinct 
vision when the visual angle for the object is enlarged in the 
ratio of the focal length of the lens to the distance of distinct 
vision. This will be plain from Fig. 126. Suppose the nearest 


Fic. 126. 


point from which the object can be observed by the naked 
eye to be the centre of the lens O. The visual angle is then 
AOB = a0b, while the visual angle for the image is a&d. 
Since these angles are always very small, we have 


a£b Oc 
aOb” Ec 


very nearly. But when AB is at a great distance, Oc is the 
focal length of the lens. By using a magnifying glass to ob- 
serve the image, the magnifying power may be still further 
increased in the ratio of the distance of distinct vision to the 
focal length of the magnifying glass. The magnifying power 
of the combination is therefore the ratio of the focal length of 
the object-glass to the focal length of the eye-glass. A con- 
cave mirror may be substituted for the object-glass for produc- 
ing the real image. 


CHAP TE Rue LIT: 
VELOCITY \OF LIGHT. 


295. Velocity Determined from Eclipses of Jupiter’s 
Moons.—Roemer, a Danish astronomer, was led to assume a 
progressive motion for light in order to explain some apparent 
irregularities in the motions of Jupiter's satellites. A few ob- 
servations of one of Jupiter’s moons are sufficient to determine 
the time of its eclipses for monthsin advance. If these observa- 
tions be made when the earth and Jupiter are on the same side 
of the sun, and the time of an eclipse occurring about six months 
later, predicted from them, be compared with the observed time 
of that eclipse, it is found that the observed time is about 162 — 
minutes later than the predicted time. This discrepancy is 
explained if it is assumed that light has a progressive motion 
and requires 162 minutes to cross the earth’s orbit, for the dis- 
tance of the earth from Jupiter in the second case is about ‘the 
diameter of its orbit greater than in the first. | 

296. Aberration of the Fixed Stars.—The apparent direc- 
tion of the light coming from a star to the earth, that is, the 
apparent direction of the star from the earth, is the resultant of | 
the motion of the light and the motion of the earth. As the 
motion of the earth changes direction the apparent direction of 
the star will change also, and the amount of that change will 
depend on the relation between the velocity of light and the 
change in the velocity of the earth in its orbit, understanding 
by change of velocity change in direction as well as in amount. 
This apparent change in the position of the stars is called aberra- 


297] VELOCITY OF LIGHT. 433 


tion. Knowing its amount corresponding toa known change 
in the earth’s motion, we may compute the velocity of light. 
This method was first employed by Bradley. 

297. Fizeau’s Method.—Several methods have been em- 
ployed for measuring the velocity of light by determining the 
time required for it to pass over a small distance on the earth's 
surface. In the form of experiment devised by Fizeau, a beam 
of light is allowed to pass out through a small hole in the shut- 
ter of a darkened room to a distant station where it is reflected 
back on itself. It returns through the opening and produces 
an image of the source. A toothed wheel is placed in front of 
the opening in such a position that, to pass out or back, the 
light must pass through the spaces between the teeth. If the 
wheel revolve slowly, as each space passes the opening in the 
shutter light will pass out, and returning from the distant sta- 
tion will enter through the space by which it made its exit. An 
image of the source will therefore be visible whenever a space 
passes the opening, and in consequence of the persistence of 
vision this image will appear continuous. Since it takes time 
for the light to go to the distant station and back, it is possible 
to give to the wheel such a.velocity that when the light which 
passed out through a given space returns, it will find the ad- 
jacent tooth covering the opening, so that no image of the 
source can be seen. If the velocity of rotation be sufficiently 
increased, the image again comes into view when the light can 
enter through the space following that by which it emerged. 
A still further increase of velocity may cause a second extinc- 
tion of the image. The experiment corisists in determining ac- 
curately the velocities for which the several extinctions and reap- 
pearances of the image occur. A high degree of accuracy can- 
not be attained because the extinction of the image is not sud- 
den. It disappears by a gradual fading away, and reappears by 
a gradual brightening. For quite a range of velocity the image 


cannot be seen at all. 
28 


A34 ELEMENTARY PHYSICS. [298 


208. Foucault’s Method.—Foucault’s method depends 
upon the use of the revolving mirror as a means of measuring a 
very small interval of time. Foucault’s experiments were re- 
peated with some modification by Michelson in 1879 and again 
in 1882. The general theory of the experiment may be under- 
stood from the following brief description. Let S (Fig. 127) be 
a narrow slit, # a mirror which may revolve about an axis in 
its own plane, Z a lens, and mw’ a second mirror. Light from 
a source behind S passes through the slit, falls on 7, is re- 
flected, when m# is in a suitable position, through the lens Z, 


Fic. 127. 


and forms an image at S’.. S and S’ are conjugate foci of the 
lens, and by so placing the lens that S shall be a little beyond 
the principal focus, S’ may be removed to as great a distance 
as desired. The mirror m’ is perpendicular to the axis of the 
lens, and at such a distance that the image S” falls upon its 
surface. It is evident that any light reflected back from m’ 
through Z will return to the conjugate focus S, whatever the po- 
sition of the mirror #’,so long as it sends the light in such a di- 
rection as to pass through Z both going and returning. If now 
the mirror m be given a rapid rotation clockwise, light passing 
through Z will return to find m in a changed position, and the 
image will be displaced from S to some point S” to the left of — 
S. Knowing the displacement SS” and the number of rotations 
of the mirror per second, the time required for light to pass 
from mm to S’and back is determined. The value of the velocity 


- 


299] VELOCITY OF LIGHT. 435 


of light, as determined by Michelson in 1879, is 299,910, and in 
1882, 299,853, kilometres per second. 

299. Influence upon the Velocity of Light, of the Motion 
of the Medium through which it Passes.—-Fizeau showed by 
experiment in 1859 that a moving transparent body increases 
or diminishes the velocity of light passing through it, not by its 
own velocity, but by a fraction of its own velocity, expressed 
by — where z is the index of refraction. This result was 
confirmed by experiments of Michelson and Morley in 1886. 
The result follows if we suppose the change of velocity of light 
in a medium to be due solely to change of density of the ether. 
Remembering that the velocity of propagation of wave motion 


in any medium is v = V2, and that the velocity ina medium 


of which the index of refraction is 7, is 7th as great as that in 


a vacuum, it may be seen at once that the density of the ether 


Fic. 128. 


in such a medium must be z’ times as great as that in a vacuum. 
In Fig. 128 let AC be a body, o/ which the index of refraction is 
z. Let the body move forward so as to occupy the position 
CD. The ether occupying the space CD and having a density 
I must in the body have a density 7’, and hence must occupy a 


; rs Sie 3 
space ZV, which is > times CD. The ether in AC must, there- 
fore, move forward through a distance CZ, while the body 


moves through a distance CD. But CE equals CD X (; Sh .) 


WI 
or CD X —=— 


n° 


Hence the ratio of the velocity of the ether 


. nm) 
to the velocity of the body is - 7 


CHAR TEN ve 
INTERFERENCE AND DIFFRACTION. 


300. Interference of Light from Two Similar Sources.— 
It has already been shown that the disturbance propagated to 
any point from a luminous wave is the algebraic.sum of the 
disturbances propagated from the various elements of the wave. 
The phenomena due to this composition of light-waves are 
called zuterference phenomena. 

Let us consider the case in which two elements only are. 


Fic. 129. 


efficient in producing the disturbance. Let A and B (Fig. 129): 
represent two elements of the same wave surface separated 
by the very small distance AB. The disturbance at m, a point 
on a distant screen 7, parallel with AB, due to these two ele- 
ments, is the resultant of the disturbances due to each sepa-. 
rately. The light is supposed to be homogeneous, and its wave 
length is represented by X. 

When the distance 2B — mA equals $d, or any odd multiple 
of 4A, there will be no disturbance at #7. Take mC = mB, and 
draw BC. mCB is an isosceles triangle; but since AB is very 


300] INTERFERENCE AND DIFFRACTION. 437 


small compared to Om, the angle at C may be taken as a right 
angle; the triangle ACS, therefore, is similar to Osm, and we 
have 


ae Ce OS 
BES an Tay VOY nearly. 


Represent sm by x, Os by c, AB by 6, AC by z X 4A, where x 
is any number. Then we have 


=F , (115) 


If z be any even whole number, the values of x given by this 
equation represent points on the screen mm at which the waves 
from A and #& meet in the same phase and unite to produce 
light. If 2 be any odd whole number, the corresponding values 
of # represent points where the waves meet in opposite phases, 
and therefore produce darkness. It appears, therefore, that 
starting from s, for which 2 =0, we shall have darkness at dis- 
tances 


and light at distances 


AGA ZAL ANE : 
O, Bp? b 5) b , CCC. 


From Eq. (115), we have 


438 ELEMENTARY PHYSICS. [300 


Since 4A is the number of wave lengths that the wave front 
from # falls behind that from A, 427, where 7 represents the 
period of one vibration, is the time that must elapse after the 
wave from A produces a certain displacement before that from 
B produces a similar displacement. The expression 


2minT 


ea 


NT 


is, therefore, the difference in epoch of the two wave systems. 
Substituting zz for e in Eq. (9g), we have 


ant sin 277 
Si lisp! maemo ea A hy J 2 cos #7)? COS (“7 — en a 
aie ee ) ve 1+ cos zz] - 


Now the intensity of light for a vibration of any given period 
is proportional to the energy of the vibratory motion. It is 
therefore proportional to the square of the maximum velocity, 
and this is proportional to the square of the amplitude. To 
find the relative intensities of light at different points, we may 
suppose Zz in the second parenthesis above to have such a value 
as shall render the cosine unity, when 


S=a(2+2cosuzj=A 


is the amplitude of the vibratory motion for any given value of 
z. Substituting for z its value and squaring, we have 


: : 20% | 
A = a2 42 cos a) 


in which A’ is proportional to the intensity of the illuminatiom 
at distances x from s. When 


300] INTERFERENCE AWD. DIFFRACTION. 439 


its cosine is I, and A* is a maximum and equal to 4a’. Asa 
increases A* diminishes, until 


264 


At= weit. whichcase A. — 0. 


A’ then increases until it becomes again a maximum, when 


26x 
Se SR —- Dite 
cx 


In short, if AB (Fig. 130) represent the line mz of Fig. 129, the 
ordinates to a sinuous curve like adc will represent the intensi- 
ties of the light along that line. 

The phenomena described above may be Pale experi- 
mentally in several ways. Young admitted sunlight into a 


5 . darkened room through a small hole 
we / NN S in a window-shutter. It fell upon 
BK B a screen in which were two small 

FIG. 130. 


holes close together, and, on passing 
through these, was received upon a second screen. Light and 
dark bands were observed upon this screen, the distances of 
which from the central band were in accordance with theory. 
Fresnel received the light from a small luminous source upon 
two mirrors making a very large angle, as in Fig. 131. The light 
reflected from each mirror proceed- 
ed as though from the image of the 
source produced by that mirror. 
The reflected light, therefore, con- 
sisted of two wave systems, from 
two precisely similar sources A and 
B. Light and dark bands were 
formed in accordance with theory. In order that the experi- 
ment may be successfully repeated reflection must take place 


A40 ELEMENTARY FH YVSICS: [301 


from the front surface of each mirror only, the angle made by 
the mirrors must be nearly 180°, and the reflecting surfaces 

must meet exactly at the vertex of the angle. 
aa Two similar sources of light may be obtained 
Ae also by sending the light through a double 
prism, as shown in Fig. 132. Light from 4 
proceeds after passing through the prism as 
from the two virtual images a and a’. 

A divided lens, Fig. 133, serves thesame purpose. The light 
from A is concentrated in two real images a and a’, from which 
proceed two wave systems as in the previous cases. What are 
really seen in these cases, when the source of light is white, are 
iris-colored bands instead of bands of light and darkness merely. 


Fic. 132. 


Fic. 133. 


When the light is monochromatic, the bands are simply alter- 
nations of light and darkness, the distances between them being 
greatest for red light, and least for blue. From Eq.(115) it ap- 
pears that, other things being equal, x varies with A, hence we 
must conclude that the greater distance between the bands in- 
dicates a greater wave length; that is; that the wave length of 
red light is greater than that of blue. 

301. Measurement of Wave Lengths.—Data may be ob- 
tained from any of the above experiments for the determination 
of the wave length of light. From Eq. (115) we have 


b 
y= oe 


ci 


302] INTERFERENCE AND DIFFRACTION. 44I 


where c, 6, and x are distances to be measured. The distance 
« is the distance from s to a point m, the centre of a light band, 
and # equals twice the number of dark bands between s and m. 
It is not necessary to consider the details of the apparatus, and 
the adjustments necessary for making these measurements. It 
is sufficient to show, in a general way, how the distance x can 
be measured. Instead of a screen, a lens or combination of 
lenses, called a positive eyepiece, is placed in the path of the 
light, and the observer looks through it towards the luminous 
source. This eyepiece has a spider-line stretched in front of it, 
which is seen magnified when the bands are observed, and lens 
and spider-line are arranged to be moved laterally by a microm- 
eter screw. By this movement the spider-line may be brought 
to coincide with the bands in succession, and the distances 
measured by the number of revolutions of the screw. Better 
methods than this of measuring wave lengths will be found de- 
scribed in § 306. 

302. Interference from Thin Films.—Thin films of trans- 
parent substances, such as the wall of a soap-bubble or a film 
of oil on water, present interference phenomena when seen in a 
strong light, due to the interference of waves reflected from the 
two surfaces of the flm. Let AA, 44 (Fig. 134) be the surfaces 
of a transparent film. Light falling on AA is partly reflected 
and partly transmitted. The reflection at the upper surface 
takes place with change of sign (§ 246). The light entering 
the film is partly reflected at the lower surface without change of 


sign, and returning partly emerges 7 
@teine upper surface. It is there a WAS : 
compounded with the wave at that , = 


moment reflected. Let us suppose Riewaees 

the light homogeneous, and the thickness of the film such that 
the time occupied by the light in going through it and return- 
ing is the time of one complete vibration. The returning wave 
will be in the same phase as the one at that moment entering, 


442 ELEMENTARY Mit YSTCS: [302 


and, therefore, opposite in phase to the wave then reflected. 
The reflected and emerging waves destroy each other, or would 
do so if their amplitudes were equal, and the result is that, ap- 
parently, no light is reflected. If the light falling on the film 
be white light, any one of its constituents will be suppressed 
when the time occupied in going through the film and returning 
is the period of one vibration, or any whole number of such 
periods, of that constituent. The remaining constituents pro- 
duce a tint which is the apparent color of the film. 

Similar phenomena are produced by the interference of that 
portion of the incident light which is transmitted directly through 
the film, with that portion which is transmitted after undergoing 
an even number of internal reflections. Since these reflections 
occur without change of sign, the thickness of the film for which 
the reflected light is a minimum is that for which the transmit- 
ted light is a maximum. 

Newton was the first to study these phenomena. He placed 
a plane glass plate upon a convex lens of long radius, and thus 
formed between the two a film of air, the thickness of which 

at any point could be determined 

a 1 when the radius of thespuemeess 
the distance from the point of con- 

tact were known. With this ar- 

rangement Newton found a black 

spot at the point of contact, and 

surrounding this, when white light 

was used, rings of different colors. 

When homogeneous light was used, 

FIG. 135. the rings were alternately light and 

dark. Let ae (Fig. 135) be the radius of the first dark ring, and 
denote it by d, and let 7 represent the radius of curvature of 
the lens. The thickness dc = ef, which may be denoted by 2, is 

ae 
Yin ope 


— 


303] INTERFERENCE: AND DIFFRACTION. 443 


Since x is very small in comparison with 27, this becomes 


Pp he 
Kael 
This distance for the first dark ring, when the incident light is 
normal to the plate, is equal to half the wave length of the light 
experimented upon. Newton found the thickness for the first 
dark ring z+<yp7 inches, which corresponds to a wave length of 
about zz} gy inches, or 0.00057 mm. This method affords a 
means of measuring the wave lengths of light, or, if the wave 
lengths be known, we may determine the thickness of a film at 
any point. 

303. Effects Produced by Narrow Apertures.—It has 
been seen (§ 274), that cutting off a portion of a light-wave by 
means of screens, thus leaving a narrow aperture for the pas- 
sage of the light, prevents the interference which confines the 
light to straight. lines, and gives rise to a luminous disturbance 
within the geometrical shadow. This phenomenon is called 
diffraction. Wet us consider the aperture perpendicular to 
the plane of the paper, and an approaching plane wave 
parallel to the plane of the aperture. Let AB (Fig. 136) 
represent the aperture, and mz one position of the approach- 
ing wave. To determine the effect 
at any point we must consider the 
elementary waves proceeding from 
the various points of the wave front 
lying between dA and &. First con- 
sider the point P on the perpendicular 
to AB at its middle point. ABZ isso 
small that the distances from P to each 
point of AB may be regarded as equal, 
or the time of passage of the light from 
each point of AB to Pmay be made 


P 


Fic, 136. 


444 BLEMEN TARY PHYSICS: [303 


the same, by placing a converging lens of proper focus between 
ABand P. Then all the elementary waves from points of AB 
meet at Pin the same phase, and the point Pis illuminated. 
Now consider a second point, P’, in an oblique 
direction from C, Fig. 137, and suppose the =m 
obliquity such that the time of passage from 
B to P’ is half a vibration period less than the 
time of passage from C to /, and a whole 
vibration period less than the time of passage 
from A tof. Plainly the elementary waves 
from Band C will meet at P in opposite phases, 
and every wave froma point between & and C 
will meet at Pa wave in the opposite phase from some point 
between Cand A. The point /’ is, therefore, not illuminated. 
Suppose another point, P” (Fig. 138), still further from. P, such 
that AB may be divided into three equal parts, each of which is 
half a wave length nearer P” than the adjacent part. It is plain 
that the two parts 4c and ca will annul each 
other’s effects at P’, but that the odd part 
Aawill furnish light. At a greater obliq- 
uity, AB may be divided into four parts, the 
distances of which from the point, taken in 
succession, differ by half a wave length. 
There being an even number of these parts, 
the sum of their effects at the point will be 
zero. Now let us suppose the point P to 
Fic. 138. occupy successively all positions to the 
right or left of the normal. While the line joining P with 
the middle of the aperture is only slightly oblique, the ele- 
mentary waves meet at FP in nearly the same phase, and 
the loss of light is small. As P approaches P’ (Fig. 137), more 
and more of the waves meet in opposite phases, the light 
grows rapidiy less, and at P’ becomes zero. Going beyond P’ 


Fic. 137. 


304] INTERFERENCE AND DIFFRACTION. 445, 


the two parts that annul each other’s effects no longer occupy 
the whole space AB, some of the points of the aperture send 
to P waves that are not neutralized, and the light reappears, 
giving a second maximum, much less than the first in intensity. 
Beyond this the light diminishes rapidly in intensity until a 
point is reached where the paths differing by half a wave length 
divide AB into four parts, when the light is again zero. Theo- 
retically, maximum and minimum values alternate in this way, 
to an indefinite distance, but the successive maxima decrease so 
rapidly that, in reality, only a few bands can be seen. 

304. Effect of a Narrow Screen in the Path of the 
_Light.—It can be shown that the effect of a narrow screen is 
the complement of that of a narrow aperture; that is, where a 
narrow aperture gives light, a screen pro- 


duces darkness. Let mm (Fig.139) be a plane 4 
wave and ABZ a surface on which the light 

falls. If no obstacle intervene, the surface |? ee 
AB will be equally illuminated. The illumi- e 

nation at any point C is the sum of the effects 

of all parts of the wave mn. Let the effects , M 


due to the part of the wave op be represented Fic. 139. 

by / and that due to all the rest of the wave by /’.. Then the 
illumination at Cis / + J’, equal to the general illumination on 
the surface. Let us now suppose mz to be a screen and fo a 
narrow aperature in it. If the illumination at C remain un- 
changed, it must be that the parts mo and gu of the wave had no 
effect, and if, for the screen with the narrow aperture, we substi- 
tute a narrow screen at of, there will be darknessat C. If, how- 
ever, a dark band fall at C when of is an aperture, a screen at of 
will not cut off the light from C. That is, if C be illuminated 
when of is an aperture, it will be in darkness when o is a 
screen, and if it be in darkness when of is an aperture, it will 
be illuminated when of is a screen. 


446 ELEMENTARY PHYSICS. [305 


305. Diffraction Gratings.—Let AB (Fig. 140) be a screen 

having several narrow rectangular apertures parallel and equi- ) 

| distant. Such a screen is called a . 

grating. Let the approaching waves, 

moving in the direction of the arrow, | 

be plane and parallel to dS. Draw 

the parallel lines ad, cd, etc., at such 

an angle that the distance from the 

centre of a to the foot of the perpen-— 

| dicular let fall from the centre of the ; 
BIG pTaS- adjacent opening on aé shall be equal 

to some definite wave length of light. It is evident that az — 

will contain an exact whole number of wave lengths, co one 

wave length less, etc. The line zz is, therefore, tangent to the 

fronts of a series of elementary waves which are in the same 

phase, and may be censidered as a plane wave, which, if it were 

received on a converging lens, would be concentrated toa focus. 

If the obliquity of the lines be increased until ae equals 2A, 

3A, etc., the result will be the same. Let us, however, suppose 

that ae is not an exact multiple of a wave-length, but some 

fractional part of a wave length, ;%9,A for example. Let 

be the fifty-first opening counting from a@; then am will be 

work X 50 = 49.54. Hence the wave from the first opening 

will be in the opposite phase to that from the fifty-first. So 

the wave from the second opening will be in the opposite phase 

to that from the fifty-second, etc. If there were one hundred 

openings in the screen, the second fifty would exactly neutralize 

the effect of the first fifty in the direction assumed. Light is 

found, therefore, only in directions given by 


Si gi mz (116) 


where 2 is a whole number, 6 the angle between the direction of 
the light and the normal to the grating, and d the distance from 


305] INTERFERENCE AND DIFFRACTION. 447 


centre to centre of the openings, usually called an element of the 
grating. Gratings are made byruling lines on glass at the rate 
of some thousands to the centimetre. The rulings may also be 
made on the polished surface of speculum metal, and the same 
effects as described above are produced by reflection from its 
surface. Since the number of lines on one of these gratings is 
several thousands, it is seen that the direction of the light is 
closely confined to the direction given by the formula, or, in 
other words, light of only one wave length is found in any one 
direction. If white light, or any light consisting of waves of 
various lengths, fall on the grating, the light corresponding to 
different wave lengths will make different angles with AC, that 
is, the light is separated into its several constituents and pro- 
duces a pure spectrum. Since different values of xz will give 
different values of 6 for each value of A, it is plain that there 
will be several spectra corresponding to the several values of 7. 
When z equals 1 the spectrum is of the frst order ; when x equals 
2 the spectrum is of the second order, etc. The grating fur- 
nishes the most accurate and at the same time the most simple 
method of determining the wave lengths of light. Knowing 
the width of an element of the grating it is only necessary to 
measure @ for any given kind of light. 

In this discussion it has been assumed that the light was 
normal to the surface of the grating. This need not be the 
case. Let AB (Fig. 141) bethe intersection with the paper ofa 
reflecting grating supposed perpen- 
dicular to it, mm an approaching 
wave front also perpendicular to 
the paper, and m’’n” the reflected 
wave front constructed as in § 278. 
The line ’’nx"’ is a tangent to all 
the elementary waves that origi- 
nate in the surface AB in conse- ae 
quence of disturbances produced by the passage of the wave 


448 ELEMENTARY PHYSICS. [305 


m'n'. Thesurface AB consists of a number of narrow, equidis- 
tant, reflecting surfaces separated by roughened channels. If the 
reflecting surfaces be considered infinitely narrow, each of them | 
will be the centre of a system of waves due to the successive in- 
cident waves similar to 72 which fall upon them. Since the 
number of the elements of the grating is finite there will be a 
finite number of such wave systems. In the diagram one of 
these systems is represented about thecentre d. Let us repre- 
sent by a, 6, c, d, etc., the centres of these systems, such that 
the distances m'’a, ab, bc, cd, etc., aré elements of the grating. 
Let us suppose the wave systems all represented, and draw 
mn’ tangent to the wave front of which the centre is a, and 


which is one wave length behind the wave to which mn” is. 


tangent. The line wz’’2’’’ will be also tangent to waves of the 
systems of which 4, c, d, etc., are the centres, and which are 
respectively two, three, four, etc., wave lengths behind the 
wave to which mm” 2’’ is tangent. These elementary waves, 
differing by successive periods, are all in the same phase, and 
mn” may, therefore, be considered as constituting a plane ~ 
wave front in which light of one particular wave length is pro- 
pagated in the direction dy. Represent by z the angle of inci- 
dence, by 7 the angle of reflection, by a the angle between 
the normal to the grating and the path of the diffracted light. 
Then z equals 7, and if m’’a equal s, the radius of the elemen- 
tary wave having its centre at a,and tangent to mn", is s sinz, 
and of the elementary wave having the same centre, and tan- 
gent to m'n'”, is s sin a. Hence, by hypothesis, we have 
ssinz—ssina=A. 

Let us designate by # the angle between the path of the in- 
cident and that of the diffracted light, and by @ the angle be- 
tween the path of the reflected and that of the diffracted light. 


If the grating be turned so that the path of the reflected light 


es 6 
coincides with dx, its normal will turn through the angle 3 and 


305] INTERFERENCE AND DIFFRACTION. 449 


DIANE 


6 
will bisect the angle 6. Hence we have z = p rie patil a So: 


Substituting these values in the equation for A we obtain 


nat 
A= 25 cos £ sin >. cane 
Hitherto the spaces from which the elementary waves pro- 
ceed have been considered infinitely narrow, so that only one 
- system of waves from each space need be considered. In prac- 
tice, these spaces must have some width, and it may happen 
that the waves from two parts of the same space may cancel 
each other. Let the openings, Fig. 142, be equal in width to 
the opaque spaces, and let the direction am be 
taken such that aeequals 2A. Then ae’ equals 
4A, or the waves from one half of each opening ‘ 
are opposite in phase to those from the other 
half, and there can be no light in the direction 
am. In general, if d equal the width of the 
opening, there will be interference and light 
will be destroyed in that direction for which Fic. 142. 


Was. are 
eo =, if the incident light be normal tothe erating. Let 
J represent the width of the opaque space. Thend+ f= s, 
and light occurs in the direction given by sin 6 = fie 
vided that the value of 6 given by this equation does not 
satisfy the first equation also. ‘ 
If d equal 7, we have 


pro- 


‘py 


elem 7X. 
ashes 


When z is even, sin 6 becomes 


sin 6 = 


SA mL an eA 2Xr 


a1 ie lage oa? 
29 : 


450 ELEMENTARY PHYSICS. [306 


and satisfies the equation 


; rX 
Pe ey eie Sebald 


a 


which expresses the condition under which light is all de- 
stroyed. Hence in this case all the spectra of even orders fail. 
Moreover, the spectra after the first are not brilliant. When f 
equals 2d the spectrum of the third order fails. 

It may be shown that whatever be the relative widths of 
the transparent and opaque spaces, one may be substituted for 
the other without altering the result. In Fig. 143 let ac rep- 
resent an opening and cd an opaque por- 
tion. Let us assume that cd equals 3ac, 
and let ad be the path of the diffracted 
light giving the spectrum of the first order; 
then we have ae = A and ae’ = 41d. Now 
let ac become the opaque portion and cd 
the opening. We will then have 74 = dA. 

Fic. 143. Each of the elementary waves from points 
between ¢ and z will be half a wave length behind a correspond- 
ing wave from some point between @ and 7, so that the waves 
coming from cz and a annul one another, and 7 is the only 
efficient portion of the opening cd. This portion 77 is equal to 
the former opening ac. Since the effect of the grating is that 
of one opening multiplied by the number of openings, it is 
plain that in this case it is indifferent whether the opentige are 
~ of the width ac or ed. 

306. Measurement of Wave Lengths.—To realize prac- 
tically the conditions assumed in the theoretical discussion of 
the last section, some accessory apparatus is required. It has 
been assumed that the wave incident upon the grating was 
plane. Such a wave would proceed from a luminous point or 
line at an infinite distance. In practice it may be obtained by 


306] INTERFERENCE AND DIFFRACTION. 451 


illuminating a very narrow slit, taking it as the source of light, 
and placing it in the principal focal plane of a well-corrected 
converging lens. The plane wave thus obtained passes through 
the grating, or is reflected from it, and is received on a second 
lens similar to the first, which gives an image either on a screen 
or in front of an eyepiece, where it is viewed by the eye. The 
general construction of the apparatus may be inferred from 
Fig. 144. It is called the spectrometer. | 
A is a tube carrying at its outer end the slit and at its inner 
end the lens, called a collimating 
lens. CJD isa horizontal graduated 
circle, at the centre of which is a 
table on which the grating is 
mounted, and so adjusted that the 
axis of the circle lies in its plane 
and parallel to its lines. In using 
a reflecting grating the collimating 
and observing telescopes may be 
fixed at a constant angle with each 
other which may be determined once for all in making the ad- 
justments of the instrument. This angle is the angle f of 
§ 305. To determine this angle the grating is turned until 
light thrown through the observing telescope upon the grating 
is reflected back on itself. The position of the graduated circle 
is then read. The difference between this reading and the 
reading when the grating is in such a position that the reflected 


a :; 5 
image of the slit is seen in the telescope is the angle z, If the 


grating be now turned until the light of which the wave length 
is required is observed, the angle through which it is turned 


6 
from its last position isthe angle. If the width of an element 


_ 


of the grating be known, these measurements substituted in 
Eq. 117 give the value of J. 


452 ELEMENTARY PHYSICS. [307° 


Wave lengths are generally given in terms of a unit called a 
tenth metre, that is, I metre X 107'*. ‘The wave lengths of 
the visible spectrum lie between 7500 and 3900 tenth metres.. 
Langley has found in the lunar radiations wave lengths as long 
as 170,000 tenth metres, and Rowland has obtained photo- 
graphs of the solar spectrum in which are lines representing 
wave lengths of about 3000 tenth metres. 

Instead of the arrangement which has been described, 
Rowland has devised a grating ruled on a concave surface, and 
is thus enabled to dispense with the collimating lens and the 
telescope. 

307. Phenomena due to Diffraction —The colors exhibited 
by mother-of-pearl are due to diffraction effects produced by 
the striated surface. Luminous rings are sometimes seen closely 
surrounding the sun or moon, due to small globules of vapor 
or particles of ice in the upper atmosphere. Similar rings may 
be seen by looking at a small luminous source through a plate 
of glass strewn with lycopodium powder. 


GEA DIVE Ren Vi, 
DISPERSION. 


308. Dispersion.—When white light falls upon a prism of 
any refracting medium, it is not only deviated from its course 
but separated into a number of colored lights, constituting an 
image called a spectrum. These merge imperceptibly from one 
into another, but there are six markedly different colors: red, 
orange, yellow, green, blue, and violet. Red is the least and 
violet the most deviated from the original course of the light. 
Newton showed by the recomposition of these colors by means 
of another prism, by a converging lens, and by causing a 
disk formed of colored sectors to revolve rapidly, that these 
colors are constituents of white light, and are separated by 
the prism because of their different refrangibilities. To arrive 
at a clear understanding of the formation of this spectrum, 
let us suppose first a small source of homogeneous light Z 
(Fig. 145). If this light fall on a converging lens from a point 


Fic: 145. 


at a distance from it a little greater than that of the prin- 
cipal focus, a distinct image of the source will be formed at the 
distant conjugate focus’ If now a prism be placed in the 
path of the light, it will, if placed so as to give the minimum 


A54 ELEMENTARY PHYSICS. [309 


deviation, merely deviate the light without interfering with the 
sharpness of the image, which will now be formed at /’ instead 
of at /Z. If the source Z give two or three kinds of light, the 
lens may: be so constructed as to produce a single sharp image 
at Zof the same color as the source, but when the prism is in- 
troduced the lights of different colors will be differently deviated 
and two or three distinct images will be found near 7’. If there 
be many such images, some may overlap, and if there be a great 
number of kinds of light varying progressively in refrangibility, 
there will be a great number of overlapping images constituting 
a continuous spectrum. 

309. Dispersive Power.—It is found that prisms of dif- 
ferent substances giving the same mean deviation of the light 
deviate the light of different colors differently, and so produce 
a longer or shorter spectrum. The ratio of the difference be- 
tween the deviations of the extremities of the spectrum to the 
mean deviation may be called the dzspersive power of the sub- 
stance. Thus if a’, ad” represent the extreme deviations, and a 


da’ as a 
the mean deviation, the dispersive power is aE 8 
sin Arx 
$ 279 we fi h (ONG 
In § 279 we find the equation Re ae and referring to 


Fig. 108 we may set sin Avx = sin (Azr+ 2dr). From the 
discussion of § 279 it appears that when the prism is in the 
position of minimum deviation, the angle Axr equals half the 


A 
refracting angle of the prism, or a and the angle xA7v equals. 


a a ; 
half the deviation, or a Hence we obtaiti 


Ad 


Sil jae 
2 


= ——; (118) 


310] DISPERSION. 455 


. A+d 

or when A is small, |) fee < ; 
“ee 

from which d= A(u—1), 


Hence we obtain 


(i es AL ie. — 1) — A(p” — NR re 
LE ss ai besa (in-m 1) in oe 


where yw’ and pw” are the refractive indices for the extreme 
colors, and yu the index for the middle of the spectrum. 

310. Achromatism.—If in Newton's experiment of recom- 
position of white light by the reversed prism the second prism 
be of higher dispersive power than the first, and of such an 
angle as to effect as far as possible the recomposition, the light 
will not be restored to its original direction, but will still be 
deviated, and we shall have deviation without dispersion. 
This is a most important fact in the construction of optical 
instruments. The dispersion of light by lenses, called chromatic 
aberration, was a serious evil in the early optical instruments, 
and Newton, who did not think it possible to prevent the dis- 
persion, was led to the construction of reflecting telescopes to 
remedy the evil. It is plain, however, from 
what has been said above, that in a combina- 
tion of two lenses of different kinds of glass, 
one converging and the other diverging, one 
may correct the dispersion of the other within 
certain limits, while the combination still acts 
as a converging lens forming real images of Fic, 146. 
objects. Fig. 146 shows how this principle is applied to the 


450 ELEMENTARY PHYSICS, [310 


correction of chromatic aberration in the object-glasses of tele- 
scopes. 

Thus far nothing has been said of the relative separation of 
the different colors of the spectrum by refraction by different 
substances. Suppose two prisms of different substances to 
have such refracting angles that the spectra produced are of the 
same length. If these two spectra be superposed, the extreme 
colors may be made to coincide, but the intermediate colors do 
not coincide at the same time for any two substances of which 
lenses can be made. Perfect achromatism by means of lenses 
of two substances is therefore impossible. In practice it is 
usual to construct an achromatic combination to superpose, not 
the extreme colors, but those that have most to do with the 

brilliancy of the image. 


The indistinctness due to chromatic aberration, existing 
even in the compound objective, may be much diminished by 
a proper disposition of the lenses of the eyepiece. Fig. 147 
shows the zegative or Huyghens eyepiece. 

Let A be the objective of a telescope or microscope. A 


point situated on the secondary axis ov would, if the objective 
were a single lens, have images on that axis, the violet, nearest 
and the red farthest from the lens. If the lens could be per- 
fectly corrected, these images would all coincide. By making 
the lens a little over-corrected, the violet may be made to fall 
beyond the red. Suppose 7 and v to be the images. #& and C 
are the two lenses of the Huyghens eyepiece. J is called the 
field-lens, and is three times the focal length of C. It is placed 


3111 DISPERSION. 457 


between the objective and its focal plane, and therefore prevents 
the formation of the images vv, but will form images at 7’v’ on 
the secondary axes o’7, o’'v. If everything is properly propor- 
tioned, 7’v’ will fall on the secondary axis o’’R of the eye-lens 
C at such relative distances as to produce ove virtual image at 
RV. It will be noted that the image 7’ is smaller than would 
have been formed by the objective. | 
The magnifying power of the in- 
strument is therefore less than it 
would be if the lens C were used 
alone as the eyepiece. This loss 
of magnifying power is more than 
counterbalanced by the increased 
distinctness. “ 

Fig. 148 shows the Ramsden or Fic. 148, 
positive eyepiece. ‘The aid it gives in correcting the residual 
errors of the objective is evident from the figure. 

311. The Rainbow.—The rainbow is due to refraction and 
dispersion of sunlight by drops of rain. The complete theory of 
the rainbow is too abstruse to be given 
here, but a partial explanation may be 
Piventa., leet. 0). Ric. vi40, representa 
drop of water, and SA the paths of the 
incident light from the sun. The light 
enters the drop, suffers refraction on 
eritrance, is reflected from. the interior 
surface near B,and emerges near C,asa 

aise wave of double curvature of which wz 
may be taken as the section. Of this wave the part near /, the 
point of inflection, gives the maximum effect at a distant point, 
and if the eye be placed in the prolongation of the line CZ per- 
pendicular to the wave surface, light will be perceived, but at 
a very little distance above or below CE& there will be darkness. 
The direction CE is very nearly that of the minimum deviation ° 


458 ELEMENTARY PHYSICS. [312 


produced by the drop with one internal reflection. It is also 
the direction in which the angle of emergence equals the angle 
of incidence. ‘The direction CA corresponds to the minimum 
deviation for only one kind of light. If this be red light, the 
yellow will be more deviated, and the blue still more. To see 
these colors the eye must be higher up, or the drop lower down. 
If the eye remain stationary, other drops below O will send to 
it the yellow and blue, and other colors of thespectrum. Since 
this effect depends only on the angle between the directions SA 
and CZ&, it is clear that a similar effect 
will be received by the eye at & from 
all drops lying on the cone swept out 
by the revolution of the line CZ and 
— all similar lines drawn to the drops 
peas above and below the drop O, about an 
axis drawn through the sun and the eye, and hence parallel to 
SA. This cone will trace out the primary rainbow having the 
red on the outer and the blue on the inner edge. The secondary 
bow, which is fainter,and appears outside the primary, is pro- 
duced by two reflections and refractions as shown in Fig. 150. 
312. The Solar Spectrum.—dAs has been seen (§ 308) solar 
light when refracted by a prism gives in general a continuous 
spectrum. Wollaston, in 1802, was the first to observe that 
when solar light is received upon a prism through a very narrow 
opening at a considerable distance, dark lines are seen crossing 
the otherwise continuous spectrum. Later, in 1814-15, Fraun- 
hofer studied these lines, and mapped about 600 0f them. That 
these may be well observed in the prismatic spectrum it is im- 
portant that the apparatus should be so constructed as to avoid 
as far as possible spherical and chromatic aberrations. The slit 
must be very narrow, so that its images may overlap as little as 
possible. The most important condition for avoiding spherical 
aberration is that the waves reaching the prism should be plane 
waves, since all others are distorted by refraction at a plane sur- 


S 


313] DISPERSION. 459 


face. Fig. 151 shows the disposition of the essential parts of the 
apparatus known as the spectroscope. Sis the slit, which may 
be considered as the source of light. C is an achromatic lens, 
called a collimating lens, so placed that S is in its principal 
focus. The wavesemerging from it will then be plane. These 
will be deviated by the prism, and the waves representing the 
different colors will be separated, so that after passing through 
the second lens Othese different colors will each give a separate 


> Niwa WE) 


FIG. 151. 


image. These images may be received upon a screen, or ob- 
served by means of an eyepiece. Sometimes a series of prisms 
is used to cause a wider separation of the different images. 

If the images at / be received on a sensitive photographic 
plate, it will be found that the image extends far beyond the 
visible spectrum in the direction of greater refrangibility, anda 
thermopile or bolometer will show that it also extends a long 
distance in the opposite direction beyond the visible red. The 
solar radiations, therefore, do not all have the power of exciting 
vision. Much the larger part of the solar beam manifests its 
existence only by other effects. It will be shown that, physi- 
cally, the various constituents into which white light is separated 
by the prism differ essentially only in wave length. 

313. Spectrum Analysis.—If, in place of sunlight, the light 
of a lamp or of any incandescent solid, such as the lime of the. 
oxyhydrogen light or the carbons of the electric lamp, illuminate 
the slit, a continuous spectrum like that produced by sunlight 
is seen, but the black lines are absent. Solids and liquids give 
in general only continuous spectra. Gases, however, when incan- 


460 ELEMENTARY PHYSICS. [313 


descent give continuous spectra only very rarely. Their spectra 
are bright lines which are distinct and separate images of the 
slit. The number and position of these lines differ with each 
gas employed. Hence, if a mixture of several gases not in 
chemical combination be heated to incandescence, the spectral 
lines belonging to each constituent, provided all be present in 
sufficient quantity, will be found in the resultant spectrum. 
Such a spectrum will therefore serve to identify.the constituents 
of a mixture of unknown composition. Many chemical com- 
pounds are decomposed into their elements, aad the elements 
are rendered gaseous at the temperature necessary for incan- 
descence. In that case the spectrum given is the combined 
spectra of the elements. A compound gas that does not suffer 
dissociation at incandescence gives its.own spectrum, which 
is, in general, totally different from the spectra of its elements. 

The appearance of a gaseous spectrum depends in some de- 
gree on the density of the gas. When the gas is sufficiently 
compressed, the lines become broader and lose their sharply 
defined edges, and if the compression be still further increased 
the lines may widen until they overlap, and form a continuous 
spectrum. Some of the dark lines of the solar spectrum are 
found to coincide in position with the bright lines of certain 
elements. This coincidence is absolute with the most perfect 
instruments at our command, and not only so, but if the bright 
lines of the element differ in brilliancy the corresponding dark 
lines of the solar spectrum differ similarly in darkness. 

The close coincidence of some of-these lines was noted as 
early as 1822 by Sir John Herschel, but the absolute coinci- 
dence was demonstrated by Kirchhoff, who also pointed out 
its significance. Placing the flame of a spirit lamp with a salted 
wick in the path of the solar beam which illuminated the slit of 
his spectroscope, Kirchhoff found the two dark lines corre- 
sponding in position to the two bright lines of sodium to be- 
come darker, that is, the flame of the lamp had absorbed from 


313] DISPERSION. 40k 


the more brilliant solar beam light of the same color as it would 
itself emit. The explanation of the dark lines of the solar 
spectrum is obvious. The light from the body of the sun gives 
a continuous spectrum like that of an incandescent solid or 
liquid. Somewhere in its course this light passes through an 
atmosphere of gases which absorbs from the solar beam such 
light as these gases would emit if they were self-luminous. 
Some of this absorption occurs in the earth’s atmosphere, but 
most of it is known to ogcur in the atmosphere of the sun 
itself. By comparison of these dark lines with the spectra of 
various incandescent substances upon which we can experiment, 
the probable constitution of the sun is inferred. 


CHAPTER Va: 
ABSORPTION AND .EMISSION. 


314. Effects of Radiant Energy.—It has been stated that 
the solar spectrum, whether produced by means of a prism or by 
a grating, may, under certain conditions, give rise to heat, light, 
or chemical changes. It was formerly supposed that these 
were due to three distinct agents emanating from the sun, giv- 
ing rise to three spectra which were partially superposed. 
Numerous experiments show, however, that, at any place in 
the spectrum where light, heat, and chemical effects are pro- 
duced, nothing which we can do will separate one of these 
effects from the others. Whatever diminishes the light at any 
part of the spectrum diminishes the heat and chemical effects 
also. Physicists are now agreed that all these phenomena are 
due to vibratory motions transmitted from the sun, which 
differ in length of wave, and which are separated by a prism, 
because waves differing in length are transmitted in the sub- 
stance of the prism with different velocities. The effect pro- 
duced at any place in the spectrum depends upon the nature 
of the surface upon which the radiations fall. On the photo- 
graphic plate they produce chemical change, on the retina the 
sensation of light, on the thermopile the effect of heat. Only 
those waves of which the wave lengths lie between 3930 and 
7600 tenth metres affect the optic nerve. Chemical changes 
and the effects of heat are produced by radiations of all wave 
lencths, 


315] ABSORPTION AND EMISSION. 463 


To produce any effect the radiations must be absorbed ; that 
is, the energy of the ethereal vibrations must be imparted to 
the substance on which they fall, and cease to exist as radiant 
energy. The most common effect of such absorption is to gen- 
erate heat, and there are some surfaces upon which heat will be 
generated by the absorption of ethereal waves of any length. 
Langley, by means of the bolometer, has been able to measure 
the energy throughout the spectrum, and has shown the exist- 
ence of lines like the Fraunhofer lines, in the invisible spectrum 
below the red.. He has demonstrated the existence, in the 
lunar spectrum, of waves as long as 170,000 tenth metres, or 
more than twenty-two times as long as the longest that can 
excite human vision. 

315. Intensity of Radiations.—The intensity of radiations 
can only be determined by their effects. If the radiations fall 
on a body by which they are completely absorbed and con- 
verted into heat, the amount of heat developed in unit time 
may be taken as the measure of the radiant energy. Let us 
suppose the radiations to emanate from a point equally in all 
directions, and represent the total intensity of the radiations by 
£. Let the point be at the centre of a hollow sphere, of which 
the radius is 7, and represent by / the intensity of the radia- 
tions per unit area of the sphere. Then, since the surface of 
the sphere equals 477’, we have 


HIS Ar iad, 
E 
and ere (119) 


_ That is, the intensity of the radiation upon a given surface 
is in the’ inverse ratio of the square of its distance from the 
source. 


464 ELEMENTARY PHYSICS. [316 


If the surface is not normal to the rays, the radiant energy 
: it receives is less, as will ap- 
pear from Fig. 152. Let ad be 
a surface the normal to which 
makes with the ray the angle 
0; then ad will receive the 
same quantity of radiant en- 
ergy as a0’, its projection on 
the plain normal to the ray. 
But ad’ equals ad cos @; and 
if 7 represent the intensity on a’d’, and J’ the intensity on ad, 
we have - 


Fic. 152. 


TT cos sd: | 


or, the intensity of the radiations falling on a given surface is 
proportional to the cosine of the angle made by the surface and 
the plane normal to the direction of the rays. 

316. Photometry.— The object of photometry is to compare 
the luminous effects of radiations. It is not supposed that the 
radiations which fall on the retina are totally absorbed by the 
nerves that impart the sensation of light. The luminous 
effects, therefore, depend on the susceptibility of these nerves, 
and can only be compared, at least when different wave lengths. 
are concerned, by means of the eye itself. The photometric 
comparison of two luminous sources is effected by so placing 
them that the illuminations produced by them respectively, 
upon two surfaces conveniently placed for observation, appear 
to the eye to be equal. If A and &’ represent the intensities. 
of the sources, / and J" the intensities of the illuminations pro- 
duced by them on surfaces at distances 7 and 7’, the ratio be- 
tween these intensities, as was seen in the last section, is 


E 
ve r Br 


Tat are a.” 
xy? 


316| ABSORPTION AND EMISSION. 465 


and when / and /’ are equal, 


Pighe = Bae 
E r 
or d | | Wer aa pert. (120) 


That is, when two luminous sources are so placed as to 
produce equal illuminations on a surface, their intensities are 
as the squares of their distances from the illuminated surfaces. 

Rumford’s photometer consists of a screen in front of which 
is an upright rod. The luminous sources are so placed that the 
rod casts two shadows near together upon the screen, and are 
adjusted at such distances that these shadows are apparently 
equal in intensity. 

In Foucault's photometer the screen is of ground glass, and 
in place of the rod a vertical partition is placed in front of and 
perpendicular to the middle of the screen. The luminous 
sources are so placed that one illuminates the screen on one 
side of the partition, and the other on the other. The parti- 
tion may be moved to or from the screen until the two illumi- 
nated portions just meet without overlapping. 

In Bunsen's photometer the sources to be compared are 
placed on the opposite sides of a paper screen, a portion of 
which has been rendered translucent by oil or paraffine. When 
this screen is illuminated upon one side only, the translucent 
portion appears darker on that side, and lighter on the other 
side; than the opaque portion. When placed between two 
luminous sources, both sides of it may, by moving it toward 
one or the other, be made to appear alike, and the translucent 
portion almost invisible. The light transmitted through this 
portion in one direction then equals that transmitted in the 
opposite direction; that is, the two surfaces are equally illumi- 
nated. ) 

30 


AG! ELEMENTARY PHYSICS. [317 


317. Transmission and Absorption of Radiations.—It is 
a familiar fact that colored glass transmits light of certain colors 
only, and the inference is easy that the other colors are ab- 
sorbed by the glass. It is oniy necessary to form.a spectrum, 
and place the colored. glass in the path of the light either 
before or after the separation of the colors, to show which 
colors are transmitted, and which absorbed. 

By the use of the thermopile or bolometer, both of which 
are sensitive to radiations of all periods of vibration, it is 
found that some bodies are apparently perfectly transparent 
to light, and opaque to the obscure radiations. Clear, white 
class is opaque to a large portion of the obscure rays of long 
wave length. Water and solution of alum are still more 
opaque to these rays, and pure ice transmits almost none of 
the radiations of which the wave lengths are longer than those 
of the visible red. Rock salt transmits well both the luminous 
and the non-luminous radiations. 

On the other hand, some substances apparently opaque 
are transparent to radiations of long wave length. A plate of 
glass or rock salt rendered opaque to light by smoking it over 
a lamp is still as transparent as before to the radiations of 
longer wave length. Selenium is opaque to light, but trans- 
parent to the radiations of longer wave length. ‘This fact ex- 
plains the change of its electrical resistance by light, but not 
by non-luminous rays. Carbon disulphide, like rock salt, 
transmits nearly equally the luminous and non-luminous rays; 
but if iodine be dissolved in it, it will at first cut off the lu- 
minous rays of shorter wave length, and as the solution be- 
comes more and more concentrated the absorption extends 
down the spectrum to the red, and finally all light is extin- 
guished, and the solution to the eye becomes opaque. The 
radiations of which the wave lengths are longer than those of 
the red still pass freely. Black vulcanite seems perfectly 
opaque, yet it also transmits radiations of long wave length. 


319] ABSORPTION AND EMISSION. 407 


If the radiations of the electric lamp be concentrated by means 
of alens,anda sheet of black vulcanite placed between the 
lamp and the lens, bodies may be still heated in the focus. 

318. Colors of Bodies.—Bodies become visible by the light 
which comes from them to the eye, and bodies which are not 
self-luminous must become visible by sending to the eye some 
portion of the light that falls on them. Of the light which 
falls on a body, part is reflected from the surface; the re- 
mainder which enters the body is, in general, partly absorbed, 
and the unabsorbed portion either goes on through the body, 
or is turned back by reflection at a greater or less depth within 
the body, and mingles with the light reflected from the sur- 
face. 

In general the surface reflection is small in amount, and the 
different colors are reflected almost in the proportion in which 
they exist in the incident light. Much the larger portion of 
the light by which a body becomes visible is turned back after 
penetrating a short distance beneath the surface, and contains 
those colors which the substance does not absorb. This deter- 
mines the color of the object. Ina few instances there seems 
to be a selective reflection from the surface. For example, the 
light reflected from gold-leaf is yellow, while that which it trans- 
mits is green. 

319. Absorption by Gases.—lIf a pure spectrum be formed 
from the white light of the electric lamp, and sodium vapor, 
obtained by heating a bit of sodium or a bead of common salt 
in the Bunsen flame, be placed in the path of the beam, two 
narrow, sharply defined dark lines will be seen to cross the 
spectrum in the exact position that would be occupied by the 
yellow lines constituting the spectrum of sodium vapor. Gases 
in general have an effect similar to that of the vapor of sodium; 
that is, they absorb from the light which passes through them 
distinct radiations corresponding to definite wave lengths, which 
are always the same as those which would be emitted by the 


468 ELEMENTARY PHYSICS. : [320 


gas were it rendered incandescent. It has been seen already 
($ 313) that the Fraunhofer lines of the solar spectrum are thus. 
accounted for. | 

320. Emission of Radiations.—Not only incandescent 
bodies, but all bodies at whatever temperature they may be, 
emit radiations. A warm body continues to grow cool until it 
arrives at the temperature of surrounding bodies, and then if 
it be moved toa place of lower temperature, it cools still further. 
To this process we can ascribe no limit, and it is necessary to- 
admit that the body will radiate heat, and so grow cooler, what~ 
ever its own temperature, if only it be warmer than surrounding 


bodies. But it cannot be supposed that a body ceases to radiate. 


heat when it comes to the temperature of surrounding bodies, 
and begins again when the temperature of these is lowered. It 
is necessary, therefore, to assume that all bodies at whatever 
temperature are radiating heat, and that, when any one of them 
arrives at a stationary temperature, it is, if no change take 
place within it involving the generation or consumption of heat, 
receiving heat as rapidly as it parts with it. This is called the 
principle of movable equilibrium of temperature. Weknow that 
if a number of bodies, none of which are generating or consum- 
ing heat otherwise than in change of temperature, be placed in 
an inclosure the walls of which are maintained at a constant 
temperature, these bodies will in time all come to the tempera- 
ture of the inclosure. It can be shown that, for this to be true,. 
the ratio of the emissive to the absorbing power must be the 
same for all bodies, not only for the sum total of all radiations, 
but for radiations of each wave length. For example, a body 
which does not absorb radiations of long wave length cannot 
emit them, otherwise, if placed in an inclosure where it could 
only receive such radiations, it would become colder than other 
bodies in the same inclosure. This is only a general statement of 
the fact which has been already stated for gases, that bodies ab- 
sorb radiations of exactly the same kind as those which they emit. 


320] ABSORPTION AND EMISSION. 469 


Since radiant energy is energy of vibratory motion, it may 
be supposed to have its origin in the vibrations of the molecules 
of the radiating. body. In § 156 it was shown that the various 
phenomena of gases are best explained by assuming a constant 
motion of their molecules. If these molecules should have 
definite periods of vibration, remaining constant for the same 
gas through wide ranges of pressure and temperature, this 
would fully explain the peculiarities of the spectra of gases. 

In § 261 it was seen that a vibrating body may communicate 
its vibrations to another body which can vibrate in the same 
period, and will lose just as much of its own energy of vibration 
as it imparts to the other body. Moreover, a body which has 
a definite period of vibration is undisturbed by bodies vibrat- 
ing in a period different from its own. This explains fully the 
selective absorption of a gas. For, if a beam of white light 
pass through a gas, there are, among the vibrations constituting 
such a beam, some which correspond in period to those of the 
molecules of the gas, and, unless the energy of vibration of 
these molecules is already too great, it will be increased at the 
expense of the vibrations of the same period in the beam of 
light. Hence, at the parts in the spectrum where light of those 
vibration periods would fall, the light will be enfeebled, and 
those parts will appear, by contrast, as dark lines. 

In solids and liquids, the molecules are so constrained in 
their movements that they do not vibrate in definite periods. 
Vibrations of all periods may exist; but if in a given case there 
were a tendency to one period of vibration more than to an- 
other, it is evident that the body would transfer to or receive 
from another, that is, it would emit or absorb, vibrations of that 
period more than of any other. Furthermore, a good radiator 
is a body so constituted as to impart to the medium around it 
the vibratory motion of its own molecules. But the same pecu- 
liarity of structure which fits it for communicating its own 
motion to the medium when its own motion is the greater, fits 


470 ELEMENTARY PHYSICS. [32x 


it also for receiving motion from the medium when its own 
motion is the less. Theory, therefore, leads us to the conclu- 
sion which experiment has established, that at.a given temper- 
ature emissive and absorbing powers have the same ratio for 
all bodies. 

321. Loss of Heat in Relation to Temperature.—The 
loss of heat by a body is the more rapid the greater the differ- 
ence of temperature between it and surrounding bodies. For 
a small difference of temperature the loss of heat is nearly pro- 
portional to this difference. This law is known as Mewton's 
law of cooling. Fora large difference of temperature the loss. 
of heat increases more rapidly than the difference of tempera- 
ture, and depends not merely upon this difference, but upon 
the absolute temperature of the surrounding bodies. An ex- 
tended series of experiments by Dulong and Petit led to a for- 
mula expressing the quantity of heat lost by a body in an in- 
closure during unit time. It is 


6 
Q = m(1.0077) (1.00777 — I), 


where @ represents the temperature of the inclosure, ¢ the dif- 
ference of temperature between the inclosure and the radiating 
body, both measured in Centigrade degrees, and # a constant 
depending on the substance, and the nature of its surface. 

322. Kind of Radiation as Dependent upon Tempera- 
ture.— When a body is heated we may feel the radiations from 
its surface long before those radiations render the body visible. 
If we continue to raise the temperature, after a time the body 
becomesred hot ; as the temperature rises still further it becomes 
yellow, and finally attains a white heat. Even this rough ob- 
servation indicates that the radiations of great wave length are 
the principal radiations at the lower temperature, and that to 
these are added shorter and shorter wave lengths as the tem- 


322] ABSORPTION AND EMISSION, 471 


perature rises. Draper showed that the spectrum of a red-hot 
body exhibits no rays of shorter wave length than the red, but 
that as the temperature rises the spectrum is extended in the 
direction of the violet, the additions occurring in the order of 
the wave lengths. At the same time the colors previously 
existing increase in brightness, indicating an increase in energy 
of the vibrations of longer wave length as those of shorter wave 
length become visible. Experiments by Nichols on the radia- 
tions from glowing platinum show that vibrations of shorter 
wave length are not altogether absent from the radiations of 
a body of comparatively low temperature, and he was led to 
believe that all wave lengths are present in the radiations from 
even the coldest bodies, but are too feeble to be detected. 
With gases, as has been seen, the radiations are apparently 
confined to a few definite wave lengths, but careful observa- 
tions of the spectra of gases show that the lines are not defined 
with absolute sharpness, but fade away, although very rapidly, 
into the dark background. In many cases the existence of ra- 
diations may be traced throughout the spectrum, and it is a ques- 
tion whether the spectra of gases are not after all continuous, 
only showing strongly marked and sharply defined maxima 
where the lines occur. In general, increase of*temperature does 
not alter the spectra of gases except to increase their intensity, 
but there are some cases in which additional lines appear as the 
temperature rises, and a few cases in which the spectrum under- 
goes a complete change at acertain temperature. This occurs 
with those compound gases which suffer dissociation at a cer- 
tain temperature, and at higher temperatures give the spectra 
of their elements. When it occurs with gases supposed to be 
elements it suggests the question whether they are not really 
compounds, the molecules of which at the high temperature 
are divided, giving new molecules of which the rates of vibra- 
tion are entirely different from those of the original body. 


472 NSHLEMEN TARY PHYSICS, [323 


323. Fluorescence and Phosphorescence.—A few sub- 
stances, such as sulphate of quinine, uranium glass, and thallene, 
have the property, when illuminated by rays of short wave 
length, even by the invisible rays beyond the violet, of emit- 
ting light of longer wave length. Such substances are fluorescent. 
The light emitted by them, and the conditions favorable to their 
luminosity, have been studied, by Stokes. It appears that the 
light emitted is of the same character, covering a considerable 
region of the spectrum, no matter what may be the incident 
light, provided this be such as to produce the effect at all. The 
light emitted is always of longer wave length than that which 
causes the luminosity. 

There is another class of substances which, after being ex- 
posed to light, will glow for some time in the dark. These are 
phosphorescent. Vhey must be carefully distinguished from 
such bodies as phosphorus and decaying wood, which glow in 
consequence of chemical action. Some phosphorescent sub- 
stances, especially the calcium sulphides, glow for several hours 
after exposure. 

324. Anomalous Dispersion.—As has been already stated, 
there is a class of bodies which show a selective absorption at 
their surfaces. The light reflected from such bodies is comple- 
mentary to the light which they can transmit. Kundt, follow- 
ing up isolated observations of other physicists, has shown 
that all such bodies give rise to an anomalous dispersion, that 
is, the order of the colors in the spectrum formed by a prism 
of one of these substances is not the same as their order in the 
diffraction spectrum or in the spectrum formed by prisms of 
substances which do not show selective absorption at their 
surfaces. Solid fuchsin, when viewed by reflected light, appears 
green. In solution, when viewed by transmitted light, it ap- 
pears red. Christiansen allowed light to pass through a prism 
formed of two glass plates making a small angle with each 
other, and containing a solution of fuchsin in alcohol. He 


324] ABSORPTION AND EMISSION, A73 


found that the green was almost totally wanting in the spec- 
trum, while the order of the other colors was different from 
that in the normal spectrum. In the spectrum of fuchsin the 
colors, in order, beginning with the one most deviated, were 
violet, red, orange, and yellow. Other substances give rise to 
anomalous dispersion in which the order of the colors is dif- 
ferent. ; 

In order to account for these phenomena, the ordinary 
theory of light is extended by the assumption that the ether 
and molecules of a body materially interact upon one another, 
so that the vibrations in a light-wave are modified by the vibra- 
tions of the molecules of a transparent body through which 
light is passing. This hypothesis, in the hands of Helmholtz 
and Ketteler, has been sufficient to account for most of the 
phenomena of light. 


CTEA TL basen 
DOUBLE REFRACTION AND POLARIZATION. 


325. Double Refraction in Iceland Spar.—lIf refraction 
take place in a medium which is not isotropic, as has been 
assumed in the previous discussion of refraction, but eolo- 
tropic, a new class of phenomena arises. Iceland spar is an 
eolotropic medium by the use of which the phenomena re- 
ferred to are strikingly exhibited. Crystals of Iceland spar 
are rhombohedral in form, and acrystal may be a perfect rhom- 
bohedron with six equal plane faces, each of which is a rhombus. 


Fic. 153. 


Fig. 153 represents such a crystal. At 4 and X are two solid 
angles formed by the obtuse angles of three plane faces. The 
line through A making equal angles with the three edges AB, 
AE, AD, or any line parallel to it, is an optzc axzs of the crys- 
tal. 


326] DOUBLE REFRACTION AND POLARIZATION. 475 


Any plane normal to a surface of the crystal and parallel to 
the optic axis is called a principal plane. If such a crystal be 
laid upon a printed page, the lines of print will, in general, ap- 
pear double. If adot be made ona blank paper, and the crys- 
tal placed upon it, two images of the dot areseen. If the crystal 
be revolved about an axis perpendicular to the paper, one of 
the images remains stationary, and the other revolves around 
it. The images lie ina plane perpendicular to the paper, and 
parallel to the line joining the two obtuse angles of the face by 
which the light enters or emerges. The entering and emerging 
light is supposed in this case to be normal to the surfaces of 
the crystal. If the crystal be turned with its faces oblique to 
the light, the line joining the images will, in certain cases, not 
lie parallel to the line joining the obtuse angles of the faces. If 
the distances of the two images from the observer be carefully 
noticed it will be seen that the stationary one appears nearer 
than the other. If the obtuse angles A and X be cut away, 
and the new surfaces thus formed at right angles to the optic 
axis be polished, images seen perpendicularly through these 
faces do notappear double. Bycutting the crystals into prisms 
in various ways its indices of refraction may be measured. It 
is found that, of the two beams into which light is, in general, 
divided inthe crystal, one obeys the ordinary laws of refraction, 
and has a refractive index 1.658. It iscalled the ordinary ray. 
The other has no constant refractive index, does not in general 
lie in the normal plane containing the incident ray, and refrac- 
tion may occur when the incidence is normal. It is the extra- 
ordinary ray. he ratio between the sines of the angles of in- 
cidence and refraction varies, for the Fraunhofer line D, from 
1.658, the ordinary index, to 1.486. This minimum value is 
called the extraordinary index. 

326. Explanation of Double Refraction.—In § 279 it was 
seen that the index of refraction of a substance is the reciprocal 
of the ratio of the velocity of light in the substance to its 


476 ELEMENTARY PHYSICS. [327 


velocity in a vacuum. It is plain, then, that the velocity of 
light for the ordinary ray of the last section is the same for all 
directions, and, if light emanate from a point within the crystal, 
the light, following the ordinary laws of refraction, must proceed 
in spherical waves about that point as a centre, as in any single 
refracting medium. The phenomena presented by the extra- 
ordinary light in Iceland spar are fully explained by assuming 
that the velocities in different directions in the crystal are such 
as to give a wave front in the form of a flattened spheroid, of 
which the polar diameter, parallel to the optic axis, is equal to 
the diameter of the ordinary spherical wave, and the equatorial 


diameter is to its polar diameter as 1.658 is to 1.486. From 
these two wave surfaces the path of the light may easily be de- 
termined by construction by methods already explained in 
$ 279, and exemplified in Fig. 154, in which ze represents the di- 
rection of the incident light, and co and ce the ordinary and 
extraordinary rays respectively. 

327. Polarization of the Doubly Refracted Light.—lIf a 
second crystal be placed in front of the first in any of the ex- 
periments described in the last section, there will be seen in 
general four images instead of two; but if the second crystal be 
turned, the images change in brightness, and for four positions 
of the second crystal, when its principal plane is parallel or at 


327] DOUBLE REFRACTION AND POLARIZATION. 477 


right angles to the principal plane of the first, two of the images 
are invisible, and the other two are at a maximum brightness. 
If one of the beams of light produced by the first crystal be 
intercepted by a screen, and the other allowed to pass alone 
through the second crystal, the phenomena presented are easily 
followed. If the principal planes of the two crystals coincide, 
only oneimage is seen. If the second crystal be now rotated 
about the beam of light as an axis, a second image at once ap- 
pears, at first very faint, but increasing inbrightness. The origi- 
nal image at the same time diminishes in brightness, and the 
two are equally bright when the angle between the principal 
planes is 45°. If the angle be 90° the first image disappears, 
and the second is at its maximum brilliancy. As the rotation is. 
continued the first image reappears, while the second grows dim 
and disappears when the angle between the principal planes is. 
180°. These changes show that the light which emerges from 
the first crystal of spar is not ordinary light. Another experi- 
ment showsthis in astill more striking manner. Letthe extra- 
ordinary ray be cut off by a screen, and the ordinary ray be 
received on a plane unsilvered glass at an angle of incidence of 
57. When the plane of incidence coincides with the principal 
plane of the spar, the light is reflected like ordinary light. If 
the mirror be now turned about the incident ray as an axis, 
that is, so turned that, while the angle of incidence remains. 
unchanged, the plane of incidence makes successively all pos- 
sible angles with the principal plane of the crystal, the re- 
flected light gradually diminishes in brightness, and when the 
angle between the plane of incidence and the principal plane 
of the crystal is 90° it fails altogether. If the rotation be con- 
tinued it gradually returns to its original brightness, which 1t 
attains when the angle between the same planes is 180°, and 
then diminishes until it fails when the angle is 270°. The ex-. 
traordinary ray presents the same phenomena except that the 
reflected light is brightest when the angle between the planes is 


478 ELEMENTARY PHYSICS. [327 


90° and 270°, and fails when that angle is 0° and 180°. Beams 
of light after double refraction present different properties on 
different sides, and are said to be polarized. The explanation 
must, of course, be found in the character of the vibratory 
motion. 

In the polarized beam it is plain that the vibrations must 
be transverse; for if the light were the result of longitudinal 
vibrations, or even of vibrations having a longitudinal com- 
ponent, it could not be completely extinguished for certain 
astmuths of the second crystal or of the glass reflector. The 
difference between ordinary and polarized light is explained if 
we assume that in both the vibrations of the ether particles 
take place at right angles to the line of propagation of the 
wave, and that in ordinary light they occur successively in all 
azimuths about that line, and may be performed in ellipses or 
circles as well as in straight lines, while in polarized light they 
occur in one plane. In the ordinary ray in Iceland spar the 
vibrations are in a plane at right angles to the optic axis. In 
the extraordinary ray they are in the plane containing the optic 


E 
axisandtheray. The equationv= iB holds for transverse 


vibrations, if by & be understood the modulus of rigidity of the 
medium. If we assume that the modulus of rigidity at right 
angles to the optic axis is a minimum, and along the optic 
axis a maximum, and varies between these two directions 
according toa simple law, all the phenomena of double refrac- 
tion and polarization in the crystal are accounted for. Ifa 
crystal be cut so as to present faces parallel to the optic axis, 
and if light enter along a normal to one of these faces, the 
vibrations, which previous to entering the crystal were in all 
azimuths, are resolved in it in two directions, that of great- 
est and that of least elasticity, or parallel to and at right 
angles to the optic axis. The wave made up of vibrations 
parallel to the optic axis is propagated with the greater 


327] DOUBLE REFRACTION AND POLARIZATION. 479 


velocity. In this case the two wave fronts continue in parallel 
planes, and upon emergence constitute apparently one beam 
of light. If the incidence be oblique and in a plane at right 
angles to the principal plane, the two component vibrations 
are still parallel to and at right angles to the optic axis, but 
refraction occurs which is greater for the ray of which the 
vibrations are in the direction of least elasticity. If the inci- 
dence be oblique and in the principal plane, it is evident that 
there may be a component vibration at right angles to the 
optic axis, but the other component, since it must be at right 
angles to the ray, cannot be parallel to the optic axis, and 
therefore cannot be in the direction of greatest elasticity in 
the crystal. The second component is, however, in the direc- 
tion of greatest elasticity in the plane of vibration, which direc- 
tion is at right angles to the first component. In general, if 
a ray of light pass in any direction within the crystal, the line 
drawn at right angles to that direction and to the optic axis, 
that is, at right angles to the plane determined by the ray and 
the optic axis, is in the direction of least elasticity. One of 
the component vibrations is in that direction. A line drawn 
at right angles to the ray and in the plane formed by it and 
the optic axis is in the direction of the greatest elasticity to 
which any vibration giving rise to that ray of light can corre- 
spond. In that direction is the second component vibration. 
The two component vibrations are therefore always at right 
angles. One of the components is always at right angles to 
the optic axis, and hence in the direction of least elasticity, 
The light resulting from this component always travels with 
the same velocity whatever its direction, and hence suffers re- 
fraction en entering the crystal or emerging from it, according 
to the ordinary law for single refraction. The other component, 
being in the plane containing the ray and the optic axis and at 
right angles to the ray, may make all angles with the optic axis 
from o° when it is in the direction of maximum elasticity and is 


480 LLEMENTAR VAT Ysl Gs [328 


propagated with the greatest velocity, to 90° when it is in adirec- 
tion in which the elasticity is the same as that for the other 
component, and the entire beam is propagated as ordinary light. — 
Light for which vibrations occur in all azimuths will, on enter- 
ing the crystal, give rise to equal components, but light already 
polarized will give rise to components the intensities of which 
are determined by the law forthe resolutions of motions. When 
its own direction of vibration coincides with that of either of 
the components, the other component will be zero, and only 
when its vibrations make an angle of 45° with the compo- 
nents can these components be equal. The varying intensi- 
ties of the two beams into which a polarized beam is divided 
by a second crystal are thus explained. 
328. Polarization by Reflection.—Light reflected from a 
¢ transparent medium is found in 
general to be partially polarized, 
and for a certain angle of inci- 
y dence the polarization is perfect. 
This angle is that for which the 
reflected and refracted rays are at 
right angles. In Fig. 155 let zy 
Fic. 155. represent the surface of a trans- 
parent medium, aé the incident, dc the reflected, and dd the re- 
fracted ray. If the angle cbd = 90°, we have r-+2= 90° also; 


' sin 2 sin 2 f 
and since “# = ——, we have uw = ——.= tan z Hence the 
sin 7 COs 2 


angle of complete polarization is given by the equation tan z 
— yw. The fact embodied in this equation was discovered by 
Brewster, and is known as Brewster's law. The angle of com- 
plete polarization is called the polarizing angle. The plane of 
incidence is the plane of polarization. The vibrations of polar- 
ized light are at right angles to the plane of polarization. In 
the transmitted ray is an equal amount of polarized light the 
vibrations of which are in the plane of incidence. 


329] DOUBLE REFRACTION AND POLARIZATION. ASI 


If a beam of ordinary light traverse a transparent medium, 
in which are suspended minute solid particles, the light which 
is reflected from them is found to be partially polarized. The 
maximum polarization is found in the light reflected at right 
angles tothe beam. The plane of polarization of the polarized 
beam is the plane of the original beam and the beam which 
- reaches the eye of the observer. 

329. Polariscopes.—In experimenting with polarized light 
we need a folarizer to produce the polarized beam, and an 
analyzer to show the effects of the polarization. A piece of 
plane glass, reflecting light at the polarizing angle, is a simple 
polarizer. Double refracting crystals, if means be employed 


Fic. 156, % 


to suppress one of the beams into which the light is divided, 
are excellent polarizers. Tourmaline is a double refracting 
crystal which has the property of being more transparent to 
the extraordinary than tothe ordinaryray. By grinding plates 
of tourmaline to the proper thickness, the ordinary ray is com- 
pletely absorbed, while the extraordinary ray is transmitted. 
The best method of obtaining a polarized beam is by the use of 
a crystal of Iceland spar in which, by an ingenious device, the 
ordinary ray is suppressed, and the extraordinary transmitted. 
Fig. 156 shows how this is accomplished. AZ is a crystal of 
considerable length. It is divided along the plane AB making 
an angle of 22° with the edge AV and perpendicular to a prin- 
cipal plane of the face AC. The faces of the cut are polished 
and the two halves cemented together again by Canada balsam 
31 


482 ELEMENTARY RM SiCas [329 


in the same position as at first. In Fig. 157, which is a section 
through AC4D of Fig. 156, ad represents the direction of the 
light which is incident upon the face AC. It is separated into 
the two rays oande. Sincethe refractive index of the balsam 
is intermediate between the ordinary and extraordinary in- 
dices of the spar, and since the angle DAZ is so chosen that 


the ray o strikes the balsam at an angle of incidence greater 
than the critical angle, the ray o is totally reflected. The ray 
e,on the other hand, having a refractive index in the spar less 
than in the balsam, is not reflected, but continues through the 
crystal. A crystal of Iceland spar so treated is called a WVicol’s 
prism, or often simply . WVzcol. ~ 

The Foucault prism is simila. to the Nicol, except that the 
two halves after polishing are not cemented together, but are 
mounted with a film of air between. ‘The total reflection of 0 
now occurs at a much less angle of incidence. The section AB 
is, therefore, much less oblique, and a shorter crystal serves for 
the construction of the prism. It will be observed that the 
section AB must be so made that the angle of incidence of @ 
shall be greater, and of é less, than the corresponding critical 
angle. Since the two critical angles are nearly the same, but 
little variation in the angle of incidence of o and @é is permissi- 
ble, and the Foucault prism is, therefore, only useful for par- 
allel rays. 

A pair of Niccl’s prisms, mounted with their axis coincid- 
ing, serve as a folariscope. The first Nicol transmits a single 


330] DOUBLE REFRACTION AND POLARIZATION. 483 


beam of polarized light the vibrations of which are in the prin- 
cipal plane. When the principal plane of the second Nicol co- 
incides with that of the first this light is wholly transmitted 
through it. If the second Nicol or analyzer be turned about 
its axis, whenever its principal plane makes an angle with the 
direction of the vibrations, these are resolved into two com- 
ponents, one in and the other at right angles to the principal 
plane. The latter is reflected to one side and absorbed, and 
the former is transmitted. Asthe angle between the two prin- 
cipal planes increases, the transmitted component diminishes 
in intensity, until when this angle becomes 90° it disappears 
entirely. In this position the polarizer and analyzer are said 
to be crossed. 

330. Effects of Plates of Doubly Refracting Crystals on 
Polarized Light.—If a plate cut from a doubly refracting sur- 
face so that its faces are parallel to the optic axis, or at least 
not at right angles to it, be placed between the crossed polar- 
izer and analyzer, if the principal plane of the plate coincides 
with, or is at right angles to, the plane of vibration, no effect is 
perceived. But if the plate be rotated so that its principal 
plane makes an angle with the plane of vibration, the motion 
may be considered to be resolved into two components, one in, 
and the other at right angles to, the principal plane of the 
plate, and these two components on reaching the analyzer are 
again resolved each into two others, one in, and the other at 
right angles to, the principal plane of the analyzer. The vibra- 
tions in the principal plane of the analyzer are transmitted 
through it, and hence, in general, the introduction of the plate 
restores the light which the crossed polarizer and analyzer had 
extinguished. It is eas) to see that the restored light will be 
most intense when the principal plane of the plate makes an 
angle of 45° with the plane of vibration of the polarized ray. 

It is not to be understood that in the plate there are two 
separate beams of light, in one ct which one set of particles is 


484 ELEMENTARY PHYSICS. [33@ 


vibrating in one plane, and in the other another set in another 
plane. What really takes place is that each particle in the 
path of the light describes a path which is the resultant of the 
two components spoken of above. Let ad, Fig. 158, bea plate of 
Iceland spar, and cd@ the direction of its optic axis. Suppose the 
path of the light perpendicular to the plane of the paper, and 
ef to represent the direction of the disturbance produced by the 
aq entrance of a plane polarized wave. A motion 
u in the direction of ef is compounded of two. 
motions, one along the axis, and the other per- 
pendicular to it. In the propagation of this 
motion to the next particle, the motion in the 
¢ ® direction of the optic axis will begin a little 
Hic. 158. sooner than that at right angles because of the 
greater elasticity in the former direction, and this difference 
becomes greater as the light is propagated intothe plate. This. 
is equivalent to a change in the relative phases of two vibra- 
tions at right angles, and this causes the path of a vibrating 
particle to change from the straight line to an ellipse. The 
result is, therefore, that, when the initial disturbance has any 
direction except in or at right angles to the principal plane of 
the plate, the motion of the vibrating particles within the 
plate becomes elliptical, the ellipses changing form as the dis- 
tance from the front surface of the plate increases. It is en- 
tirely admissible, however, in the discussion of the problem to 
substitute for the actual motion its two components, as was. 
done above. Ae 
It remains to consider what is the effect of the retardation 
or change of phase of one of the components with respect to 
the other. It will be remembered that in the analyzer each ray 
from the plate is again resolved into two components, and that 
two of these components are in the principal plane of the ana- 
lyzer and are transmitted. These two componénts will evi- 
dently differ in phase just as did the two motions from which 


330] DOUBLE REFRACTION AND POLARIZATION. 485 


they were derived, and since they are in the same plane their 
resultant is represented by their algebraic sum. If they differ 
in phase by half a period their algebraic sum will be zero, 
and no light will be transmitted by the analyzer. This will 
occur for a certain thickness of the interposed plate. If the 
light experimented upon be white, it may occur for some wave 
lengths and not for others. Hence, some of the constituents 
of white light may fail in the beam transmitted by the analyzer, 
and the image of the plate will then appear colored. A study 
ofthe resolution of the vibrations for this case shows that, of 
the two beams formed in the analyzer, one contains just that 
portion of the light that the other lacks; hence if the analyzer 
be turned through go°, the image will change to the comple- 
mentary color. In Fig. 159, let ad represent the plane of the 
vibrations in the polarized ray, and let cd and ef represent 
the two planes of vibration of the rays in the in- 

terposed plate. At the instant of entering the @ 
plate, the primary vibration and its two compo- 
nents will have the relation shown in the figure. 
The two components are then in the same phase. 
As the movement penetrates the plate, one com- 
ponent falls behind the other, and the relation of 
their phases changes, until, with a retardation of 
one wave length, the phases are again as in the figure. Sup. 
pose the thickness of the plate such that this retardation occurs 
for some constituent of white light. After leaving the plate 
the relative phases of the components remain unchanged and 
the constituent in question enters the analyzer as two vibra- 
tions at right angles and in the same phase. In Fig. 160, let oe 
and od represent the two components, and #4 and yy the two 
planes of vibration inthe analyzer. oe will give the components 
om and on, and od the components om’ and on’. Since the com- 
ponents om and om’ annul one another, the color to which they 
correspond is wanting in the light resulting from vibrations in 


Fic. 159 


che plane xx, while since the components oz and on’ are added, 
this color is found in full intensity among the vibrations in the 
plane yy. For light of other wave 
lengths, the relative retardation is. 
different, but for each vibration 
period, the component in the di- 
rection #x% combined with that in 
the direction yy represents the 
total light for that period in the 
beam entering the analyzer; that 
Fic. 160. is, the total effect of vibrations in 
the direction xx combined with that of vibrations in the direc- 
tion yy must produce white light, and one effect must, there- 
fore, be the complement of the other. 
Let us suppose the plate thick enough to cause a retarda- 
tion equal to a certain number of wave lengths, which we will 


assume to be ten, of the shortest waves of the visible spec- 


trum. Since the longest waves of the visible spectrum are 
about twice the length of the shortest, they will suffer a retar- 
dation of five wave lengths. Other waves will suffer a retar- 
dation of nine, eight, seven, and six wave lengths. But, as was. 
seen above, a retardation of one or more whole wave lengths. 
of any kind of light causes extinction of that kind of light in 
the beam transmitted by the crossed analyzer. In the case 
considered the transmitted beam will lose six kinds of light 
distributed at about equal distances along the spectrum. The 
light remaining will consist of the different colors in about the 
same proportions as they exist in white light, and the beam 
will therefore be white but diminished in intensity. Hence, 
when a thick plate is interposed between the crossed polarizer 
and analyzer the restored light is white. 

331. Elliptic and Circular Polarization.—In the last sec- 
tion, in discussing the effects of a thin plate, we considered the 
two components of the vibratory motion propagated from it. It 


480 ELEMENTARY PHYSICS. [33u° 


331] DOUBLE REFRACTION AND POLARIZATION. 487 


was stated that the real motion of the vibrating particles was 
in general elliptical. Let us consider more fully the real mo- 
tion. Let us suppose that the light is light of one wave length 
only, and that, as before, the principal plane of the plate makes 
an angle of 45° with the plane of vibration of the incident light. 
In Fig. 161 let yy represent the original plane of vibration, and 
ab and cd the planes of maximum and minimum elasticity in 
the plate. As already explained, the first disturbance as the 
; light enters the plate is in the direction yy; 
but as the disturbance is propagated into 
the plate, each disturbed particle receives 
an impulse first of all in the direction cd of 
createst elasticity, then in other directions 
between cd and ad, and finally in the direc- 
tion ad. From this results an elliptical or- 
bit with the major axis in the direction yy. 
To determine this orbit exactly it is only 
necessary to take account of the time that 
elapses between the impulse in the direction 
ca and the corresponding impulse in the direction ad. It is suff- 
cient to consider any particle as actuated by two vibratory mo- 
tions in the directions cd and aé at right angles, and differing in 
phase. In Fig. 161, one side of the rect- 
angle represents the greatest displace- 
ment in the direction cad, and the other 
side the displacement occurring at the *\ © ils 
same instant in the direction da. The eK 
point 7 will represent the actual position 

of the vibrating particle. Constructing 

now the successive displacements of the 

particles in the directions cd and éa and ¢ 
combining these, we have the elliptical 

path as shown. As the light penetrates 

farther and farther into the plate the relative phases of the two 
vibrations change continually, and the ellipse passes through 


Fic. 161. 


Fic. 162. 


488 ELEMENTARY PHYSICS. 1331 


all its forms from the straight line yy to the straight line ax 
at right angles to it and back to the straight line yy. The 
direction of the path of the particle in the surface of the 
plate as the light emerges will be the direction of the path 
of all the particles in the polarized beam beyond the plate. 
If the component vibrations be in the same phase, that is, 
if they reach their elongations in the directions da and cd 
(Fig. 162) at the same instant, the resultant vibration is in 
the line yy and the light is plane polarized exactly as it 
left the polarizer. This will occur when the retardation of 
light in the plane of da with respect to that in the plane of cd 
is one, two, or more whole wave lengths. 
When the retardation is one half, three 
halves, or any odd number of half wave 2% 
lengths, the phases of the two vibrations are 
as shown in Fig. 163, and the resultant is a 
plane polarized beam the vibrations of which 
are at right angles to those of the beam from 
the polarizer. A case of special interest is 
shown in Fig. 164, in which the difference of 
phase is one fourth a period, and the result- Fic. 163. 
ant vibration isa circle. A difference of three fourths will give 
a circle also, but with the rotation in the opposite direction. 
A plate of such thickness as to produce a 
retardation of one quarter of a wave 
a length will give a circular vibration, and 
a se the beam issuing from the plate is then 
oe ; circularly polarized. Its peculiarity is that 
the two beams into which it is divided by 
a Ne a double refracting crystal are always of 
pre ie the same intensity, and no form of ana- 
fo ‘., lyzer will distinguish it from ordinary 
light. Quarter wave plates are often made 
by splitting sheets of mica until the re- 
Fic. 164. quired thickness is obtained. 


Ory 
. 


y 


334] DOUBLE REFRACTION AND POLARIZATION. 489 


332. Circular Polarization by Reflection.—It has been 
seen that light reflected from a transparent medium at a cer- 
tain angle is polarized, and that an equal amount of polarized 
light exists in the refracted beam. Light totally reflected in 
the interior of a medium is also polarized, and here, there being 
no refracted beam, the two components exist in the reflected 
light, but so related in phase that the light is elliptically polar- 
ized. Fresnel has devised an apparatus known as Fresnel’s 
rhomb, by means of which circularly polarized light is obtained 
by two internal reflections of a beam of light previously polar- 
ized in a plane at an angle of 45° with the plane of incidence. 

333. Effect of Plates Cut Perpendicularly to the Axis 
from a Uniaxial Crystal.—A crystal, such as Iceland spar, 
which has but one optic axis, is called a uzzaxzal crystal. Polar- 
ized light passing perpendicularly through a plate cut from such 
a crystal perpendicularly to its optic axis suffers no change. If, 
however, the plate between the crossed polarizer and analyzer 
be inclined to the direction of the beam, light passes througch 
the analyzer. It is generally colored, the color changing with 
the obliquity of the plate. If a system of lenses be used to 
convert the polarized beam into a conical pencil and the plate 
be placed in this perpendicular to its axis, the central ray of 
the pencil will be unchanged, but the oblique rays will be 
resolved except in and at right angles to the plane of vibration, 
and there will appear beyond the analyzer a system of colored 
rings surrounding a dark centre, and intersected by a black 
cross. If the analyzer be turned through 90°, a figure comple- 
mentary to the first in all its shades and tints is obtained: the 
black cross and centre become white, and the rings change to 
complementary colors. ! 

334. Biaxial Crystals.—Most crystals have two optic axes 
or lines of no double refraction, instead of one. They are 
biaxial crystals. Their optic axes may be inclined to each 
other at any angle from 0° to 180°. The wave surfaces within 


490 ELEMENTARY PHYSICS. [335 


these crystals are no longer the sphere and the ellipsoid, but 
surfaces of the fourth order with two nappes tangent to each 
other at four points where they are pierced by the optic axes. 
Neither of the two rays in sucha crystal follows the law of or- 
dinary refraction. The outer wave surface around one of the 
points of tangency has a depression something like that of an 
apple around the stem. By reference to the method already 
employed for constructing a wave front, it will be seen that there 
may be such a position for the incident wave that, when the 
elementary wave surfaces are constructed, the resultant wave 
will be a tangent to them in the circle around one of these de- 
pressions where it is pierced by the optic axis. Now since the 
direction of a ray of light is from the centre of an elementary 
wave surface to the point of tangency of that surface and the 
resultant wave, we shall have in this case an infinite number of 
rays forming a cone, of which the base is the circle of tangency. 
in other words, one ray entering the plate in a propér direction 
may be resolved into an infinite number of rays forming a cone, 
which will become a hollow cylinder of light on emerging from 
the crystal. VThis phenomenon is called conical refraction. It 
was predicted by Hamilton from a mathematical analysis of 
the wave propagation in such crystals. 

If a plate be cut from a biaxial crystal perpendicular to the 
line bisecting the angle formed by the optic axes, and placed 
between the polarizer and analyzer in a conical pencil of light, 
there will be seen a series of colored curves called lemniscates, 
resembling somewhat a figure 8. -The existence of this phe 
nomenon was also predicted and the forms of the curves in- 
vestigated by mathematical analysis before they were seen. 

335. Double Refraction by Isotropic Substances when 
Strained.—A piece of glass between the crossed polarizer and 
analyzer, if subjected to forces tending to distort it, will restore 
the light beyond the analyzer and in some cases produce 
chromatic effects. Unequal heating produces this result, and 


336] DOUBLE REFRACTION AND POLARIZATION. 49t 


a long tube made to vibrate longitudinally shows it when the 
light crosses it near the node. Pieces cut from plates of un- 
annealed glass exhibit double refraction when examined by 
polarized light. Indeed, the absence of double refraction is a 
test of perfect annealing. 

336. Effects of Plates of Quartz.—A quartz crystal is uni- 
axial, and gives an ordinary and an extraordinary ray, but is un- 
like Iceland spar in that the extraordinary wave front in it isa 
prolate spheroid and lies wholly within the spherical ordinary 
wave, not touching it even where it is pierced by the optic 
axis. The effects due to plates of quartz in polarized light 
differ very greatly from those due to Iceland spar or selenite. 
If a plate of quartz cut perpendicular to the axis be placed in 
a beam of parallel, homogeneous, plane polarized light at right 
angles to its path, the light is, in general, restored beyond the 
analyzer, and is unchanged by the rotation of the quartz 
through any azimuth. If the analyzer be rotated through a 
certain angle, depending on the thickness of the quartz plate, 
the light is extinguished. It is evident that the plane of polar- 
ization has simply been rotated through a certain angle. Light 
of a different wave length would have been rotated through a 
different angle. A beam of white polarized light, therefore, 
has the planes of polarization of its constituents rotated through 
different angles, and the effect of rotating the analyzer is to 
quench one after another of the colors as the plane of polari- 
zation for eachis reached. The result is a colored beam which 
changes its tint continuously as the analyzer rotates. 

The best explanation of these phenomena was given by 
Fresnel. It is found that neither of the two beams from a 
quartz crystal is plane polarized. The polarization is in gen- 
eral elliptical, but becomes circular for waves perpendicular 
to the axis of the crystal, the motion in one ray being right- 
handed and in the other left-handed. Each particle of ether in 
the path of the light within the crystal is actuated at the same 


492 ELEMENTARY PHYSICS. [337 


time by two circular motions in opposite directions. Its real 
motion is in the diameter which bisects the 
chord joining any two simultaneous hypo- 
thetical positions of the particle in the two 
circles. In Fig. 165 let Pand\Oerepsegea. 
these two simultaneous positions. It is 
plain that the two components in the direc- 
tion AB have the same value and are added, 
c while those at right angles to AB are equal 
Bee toa. and opposite and annuleach other. So long 
as the two components retain the same relation as that assumed, 
the real motion of the particle is in the ine AS. But in the 
quartz plate one of the motions is propagated more rapidly 
than the other, and another particle farther on in the path of 
the light may reach the point P in one of its circular vibra- 
tions at the same time that it reaches Q' in the other. This 
will give CD as its real path, and the plane of its vibration has 
been rotated through the angle BOD. When the light finally 
emerges from the plate its plane of vibration will have been 
rotated through an angle which is proportional to the thick- 
ness of the plate and depends upon the wave length of the 
light employed. A plate of quartz one millimetre in thick- 
ness rotates the plane of polarization of red light corresponding 
to Fraunhofer’s line B, 15° 18’, of blue light corresponding to 
the line G, 42° 12’. Some spetimens of quartz rotate the 
plane of polarization in one direction, and some in the oppo- 
site. Rotation which is related to-the direction of the light as 
the directions of rotation and propulsion in a right-handed 
screw is said to be right-handed, and that in the opposite 
direction 1s left-handed. | 
337. Artificial Quartzes.—Reusch has reproduced all the 
effects of quartz plates by superposing thin films of mica, 
each film being turned so that its principal plane makes an 
angle of 45° or 60°, always in the same direction, with that of 


339] DOUBLE REFRACTION AND POLARIZATION. 493 


the film below. If a plane polarized wave enter such a com- 
bination, an analysis of the resolution of the vibration as it 
passes from film to film will show that the result is equivalent 
to that of two contrary circular vibrations, one of which 
is propagated less rapidly than the other. This helps 
to establish Fresnel’s theory of the rotational effects of 
quartz. 

338. Rotation of the Plane of Polarization by Liquids.— 
Many liquids rotate the plane of polarization, but toaless amount 
than quartz. A solution of sugar produces a rotation varying 
with the strength of the solution, and instruments called sac- 
charimeters are made for determining the strength of sugar solu- 
tions from their effect in rotating the plane of polarization. In 
these instruments the effect is often measured by interposing a 
wedge-shaped piece of quartz, and moving it until a thickness 
is found which exactly compensates the rotation produced by 
the solution. 

339. Electromagnetic Rotation.—Faraday discovered that 
when polarized light passes through certain substances in a 
magnetic field, the plane of polarization is rotated through a 
certain angle. The experiment succeeds best with a very dense 
glass consisting of borate of lead, so placed that the light may 
. traverse it along the lines of magnetic force, in the field pro- 
duced by a powerful electromagnet. The amount of rotation 
is proportional to the difference of magnetic potential between 
the two ends of the glass. The direction of rotation, as was. 
shown by Verdet, is generally right-handed in diamagnetic 
media, and left-handed in paramagnetic media. It also depends 
upon the direction of the lines of force, and is therefore re- 
versed by reversing the current in the electromagnet. It fol- 
lows, also, that if the light, after traversing the glass with the 
lines of force, be reflected back through the glass against the 
lines of force, the rotation will be doubled. It is important to. 
note that this is the reverse of the effect produced by quartz, 


494 ELEMENTARY PHYSICS. [339 


solutions of sugar, etc., which rotate the plane of polarization 
in consequence of their own molecular state. When light of 
which the plane of polarization has been rotated by passage 
through such substances is reflected back upon itself, the rota- 
tion produced during the first passage is exactly reversed during 
the return, and the returning light is found to be polarized in 
the same plane as at first. 

In the magnetic field the effect is as though the medium 
which conveys the light were to rotate around an axis parallel 
to the lines of force, and to carry with it the plane of vibration. 
Evidently the plane of vibration would be turned through a 
certain angle during the passage of the light through the body, 
and would be turned still further in the same direction if the 
light were to return. An illustration may be drawn from the 
movement of a boat rowed across a current. If we row at right 
angles to the current, the boat is carried downward, and lands 
on the opposite shore below the point of starting. If then 
we row back, still at right angles to the current, the boat on 
reaching the shore from which it started is farther down the 
stream. On the other hand, in moving across a still lake, we 
might find ourselves compelled to take an oblique course on 
account of rocks or other permanent obstacles. Ifso, we should, 
on returning,be compelled to retrace our path, and would land 
at the point of starting. 

When we remember that iron becomes magnetic by the 
effect of currents of electricity flowing in conductors around it, 
and that Ampére conceived that a permanent magnet consists 
of molecules surrounded by electric currents, all in the same 
direction, it is easy to imagine that the magnetic field isa re- 
scion where the ether is actuated by vortical motions, all in the 
same direction, and in planes at right angles to the lines of 
magnetic force. Such a motion would account for the rota- 
tional effects of the magnetic field upon polarized light. 

Not only glass but most liquids and gases exhibit rotational 


340] _ DOUBLE REFRACTION AND POLARIZATION. 495 


effects when placed in a powerful magnetic field; and Kerr has 
shown that when light is reflected from the polished pole of 
an electromagnet, its primitive plane of polarization is rotated 
when the current is passed, in one direction fora north pole, 
and in the opposite direction for a south pole. 

340. Maxwell’s Electromagnetic Theory of Light.—In 
Maxwell's treatment of electricity and magnetism, he assumed 
that electrical and magnetic actions take place through a uni- 
versal medium. In order to determine whether this medium 
may not be identical with the luminiferous ether, he investigated 
its properties when a periodic electromagnetic disturbance is 
supposed to be set up in it, such as would result from a rapid 
reversal of electromotive force at a point, and compared them 
with the observed properties of the ether, on the assumption 
that light is an electromagnetic disturbance. He showed that 
such a disturbance would be propagated through the medium 
in a way similar to that in which vibrations are transmitted in 
an elastic solid. He showed further that if light were such a 
disturbance, its velocity in the ether should be equal to v, the 
ratio of the electrostatic to the electromagnetic system of units. 
Numerous measurements of the velocity of light and of this 
ratio show that they are very nearly equal. 

He also showed that the indices of refraction of transparent 
media should be equal to the square roots of their specific in- 
ductive capacities. This relation may be’ deduced as follows: 
We may suppose electrical and luminous effects to be trans- 
mitted through the dielectric by means of the ether within it, 
and farther suppose electrical effects in the medium, and there- 
fore its specific inductive capacity, to be proportional to dis- 
placements produced in the ether in it by electrical forces. 
Other things being equal, a displacement is inversely propor- 
tional to the elasticity of the medium. The velocity of propa- 
gation of a disturbance is directly proportional to the square 
root of the elasticity, if the density of the ether remain constant, 


496 ELEMENTARY PHYSICS. [340 


and the index of refraction for light is inversely as the velocity 
of propagation. Hence the index of refraction is equal to the 
square root of the specific inductive capacity. ‘To illustrate 
this let us suppose the specific inductive capacity of a dielectric 
to be 2. This means that a given electric force produces in the 
ether in that substance twice the displacement which it would 
produce in the ether in air. Hence the elasticity of the ether 
in that substance is one half as great as in air, the velocity of i 
propagation of light in it will be to the velocity in air as 1: /2, j 
and the index of refraction will be 2. 

Measurements of indices of refraction and specific inductive 
capacities have shown that the relation which has been stated . 
holds true in many cases. Hopkinson has shown, however, 
that it does not hold true for animal and vegetable oils. 

The theory leads to the conclusion that the direction of pro- 
pagation of the electrical disturbance and the accompanying 
magnetic disturbance at right angles to it is normal to the plane 
of these disturbances. By making the assumption, which is 
justified by Boltzmann’s measurements upon sulphur, that an 
eolotropic medium has different specific inductive capacities in 
different directions, Maxwell shows also that the propagation 
of the electrical disturbance in a crystal will be similar to that 
of light. It has also been shown that the electrical disturbance 
will be reflected, refracted, and polarized at a surface separating 
two dielectrics. 

Lastly, Maxwell concludes that, if his theory be true, bodies 
which are transparent to the vibrations of the ether should be 
dielectrics, while opaque bodies should be good conductors. 
In the former the electrical disturbance is propagated without 
loss of energy; in the latter the disturbance sets up electrical 
currents, which heat the body, and the disturbance is not pro- 
pagated through the body. Observation shows that, in fact, 
solid dielectrics are transparent, and solid conductors are opaque, 
to radiations in the ether. Maxwell explains the fact that 


340] DOUBLE REFRACTION AND POLARIZATION. 497 


many electrolytes are transparent and yet are good conductors 
by supposing that the rapidly alternating electromotive forces 
which occur during the transmission of the electrical disturbance 
act for so short a time in one direction, that no complete sepa- 
ration of the molecules of the electrolyte is effected. No elec- 
trical current, therefore, is set up in the electrolyte, and elec- 
trical energy is not lost during the transmission of the disturb- 
ance. : 
29 


TABLES. 


TABLE I. 


Units oF LENGTH. 


Foot 
Inch 


Pound 
Grain 


30.48 cm. 
2.54 cm. 


Units oF MAss. 


453-59 grams, 
0.0648 grams. 


TABLE WII. 


log. 1.484015 
log. 0.404830 


log. 2.656664 
log. 8.811575 


ACCELERATION OF GRAVITY, 


g = 980.6056 — 2.5028 cos 2/— o.000003%, where / is the latitude 
and / its height in centimetres above the sea level. 


g at Washington = 
g at New York = 


Kilogram-metre 
Foot-pound 


II 


UNITS 


Watt 
Horse-power 


100, 
13,825¢ ergs. 
1.355 X 107 ergs, log 7.13200, when ¢ = 980. 


980.07 | g at Paris 


980.26 | g at Greenwich 


TABLE III. 


UNITS OF WORK. 
000g’ ergs. ~ 


OF RATE OF WORK 


i ll 


ING. 


= 10’ ergs per second. 


= 550 foot-pounds per second. 


= 746 Watts. 


UNIT OF HEAT, 


Lesser calorie (gram-degree) = 4.16 X 107 ergs. 


"y 


of the statior ¥ 
an 


\ 


TABLES. * 499 


TABLE IV. 
DENSITIES OF SUBSTANCES AT 0°, 


The densities of solids given in this table must be taken as only approx- 
imate. Specimens of the same substance differ among themselves to such an 
extent as to render it impossible to give more precise values. 


HOS aS a ave We LTOU WLOURNIL). 66 ccs aye so 7G. teres 
Meee Ok i eee ee oftio ate TEA PMMALRSCT 5 ate. ei tebe aia ey Tae iLOn hay, 
RJOPPers. vee es Pal ses spoa'. oy5ik 6 Geter ete efit salaces beg ee tees Setpgtior sh i Le. 
Mes Cieme ts Geeks chk Doe Gens ns DOr te DIRTEDLY te cadS scans sees ee okt SOO 
TFIASE CET OWE) of o%70 55 85. Fea ater Pte PamUnUlils als gy's's ya histoce a rec mes ya ae 
fayaromwen).. si... Wa aie eu 6 0.0000895 | Silver...... Ce EOIN. 10.5 
Da werceeereccccverenes2O.QtS | ZiMCescercessssetenecreeeee ae gS ke 

TABLE V. 

UNITS OF PRESSURE FOR g = O81. 
Grams per sq. cm. Dynes per sq. cm, 

Pound per square inch. .........eese0e 70.31 6.9 ao 
TMC OLUNETCULY At O°. oc ecrecesesece 34.534 3.388 X 104 
I millimetre of mercury at 0°......... 1.3596 1333.8 
Peete Uere (FOO MM.) .so.cceeciesces 1033.3 1.0136 X 108 
I atmosphere (30 inches).......... aes 1036. 1.0163 X 108 

TABLE VI. 


Hes BCT Ys 


If p is the force in dynes per unit area tending to extend or compress a 


d 
body, the linear elasticity is a , and the volume elasticity is i 
y yea YS Ww 
dp. dp 
dl dv 
PASE faye ois ote ae 6.03 X 10!! 4.15 X to! 
BRP O AS G00 ac a aia oie phn FW 408 hoy oe Shas Ode KATOM 
PGE es ns asca nv. e 1207 10? eh 
Mercury.......08: al Pas 3.44) <.10" 


REY Pe. Gis sso Bd o. ae 2.02 * 10! 


500 


ELEMENTARY PHYSICS. 


TABLE VII. 


ABSOLUTE DENSITY OF WATER AT 7° IN GRAMS PER CUBIC CENTIMETRE. 


Zo: Density. 
0.999884 
0.999941 


0.999982 
I .000004 


I.000013 
I .000003 


0.999983 


OURS cenit. tes hs 


ania steele es 


DENSITY OF 


eon Density. 
13.5953 
13.5707 


Ov.ccccscescrver 


TO, cccccccccee 


e: Density. 
Thick te a eeeee 0.999946 
By cestahe wim tauiee. ate 0.999899 
mi ajeisie awe eye 0.999837 
LQ iviaiieiste « Sam 2) 4s 0.999760 
5 ais a's suki seas 0.999173 
2D. sininis clans aly 0.998272 


BO. wiecnsceces 0.995778 


TABLE VIII. 


TABLE IX. 


ag Density. _ { 
AO. a oon svees 0.09230 
BO. vi. teens - 0.98521) ae 
L618 Maer ap o o« OF 095 3G5 
704 ss eae 0.97795. 
SO. sis bean sce « OsQ7ae 
QO. 06 s's nas meen) en 


TOO. . cesses wee OnGan am 


MERCURY AT 7°, WATER AT 4° BEING I. 
log. ica Density. log. 7 & 
1.13330. 1 (20.5 < asa sine es eee een 1.131826 
T.532607)] 30ie ns sen ic ple OL it 1.13103. 


COEFFICIENTS OF LINEAR EXPANSION, 


Temperature. a= =m 
AIUMININMN Ts'o pee cee ics ses LO PLO EL OUR 0.0000235 
Brasstn cies cheba esi ave ose Oo to 100 0.0000188. 
Coppers aisha 5 Skis ate iiarooe o to. 100 0.0000167 
German silver...... piniee Vin © to 100 0.0000184. 
(F1aSS. Salat: sive sept a adeieeradots Oo to 100 O.000007I 
LOM y sient cle ene sa hipheteis Cn tee mena Len 0.00001 23 
Lead ie aiken eetpe'e esiva = leven’ O47 tO a 100 0.0000280. 
Platinum ax -aus:3 Ghul emiate ett © to 100 0.0000089, 
Silverpiete. My Ay ee weenie o to 100 0.0000194 
ZIDCH chil Ras kee Aer 3, o to 100 0.0000230: 

é : es 
Coefficients of voluminal expansion, —- = 3a. 


Gae 


TABLES. 501 


TABLE X. 
SpecIFIC HEATS—WATER AT O° =I, 


Solids and Liquids. 


SMOMTIITHIEING S's oa vibhsicce ce tees ROE SM NLSECUEV ch ols eres 66 0. 4/00's'« wie e's 0.033 
Pema ee MIG ieiss Ape ode 6 ss 0's DONE Hk ORIN Ge de ok eiee era eee 0.032 
PGC Hetee vay a's Ghee ec ees se REPONY SIAN LECT ou o cagstaie oa picielas so Ke a. 0.056 
DRM R ss ek sses Se hss Golies) Water (oO TO 100"). 4.25. e se 1.005 
PSR rrateettre co a's" 3c Se OVO RAMAN acs ease ale secre 'e keels aise t's 0.056 


Gases and Vapors at Constant Pressure. 


tee. oe ier pes ge ME NTTOGET nr socle 5 Behe s eae s 0.244 
So! a oe ra FiA LO ORV OCT yale sales ale ae eek seen OL2E7 


TABLE AL 


I. MELTING Pornts. II. Bortinc Points. ILI. HEATs oF LIQUEFACTION. 
IV. HEATS OF VAPORIZATION. V. MAXIMUM PRESSURE OF VAPOR AT 0° 
IN MILLIMETRES OF MERCURY. 


a3, II. III. IV. V. 

LTE Sy i i: — 33.7" 2 2904 at7.8° 3344 
Carbon dioxide ........ — 65° — 78.2 aye 49-3 at.O° 27100 
PE ICANGe es clnv eee soos es <t — 33.6 ere ry 4560 
BPOUDGE adc as rc ss egiées es 1200 

VT pe ey Cole ae 325 a: Bat. ee 
UEMUUEN ov nies cr eccas ese — 39 357 2.8 62 0.02 
Nitrous oxide, N2O..... aH — 105 oe oP 24320 
PUESEETD oe py oo s.0 0s cle oes 1780 ae Ay Pe 

SOE gx oss 00 0 00 5 Keye\e) vie 21.1 ae dig 
PETES. dice sees ste fe) 100 80 537 4.6 


Es eas niece 2 oe 415 we Pree | 


502 ELEMENTARY PHYSICS. 
TABLE XII. 


MAXIMUM PRESSURE OF VAPOR OF WATER AT VARIOUS TEMPERATURES IN 
(I.) DYNES PER SQUARE CENTIMETRE, (II.) MILLIMETRES OF MERCURY. 


Temp. Hy ie Temp. ic If, 
er TOL erste lah tats 1236 es BG oS a'eeaten 1.985 X 108 149. 
TO tee ete 2790 i Te feo gre 4.729 X 10° 355~ 

Opry eeunts an gee OT aa ASG 4 IG pase eae 10/14 eee 760. 
AO ee sh yp Vinten 12220 6721 2126 ed ara 19.88 x 10° “Td4qme 
Bums BG ido has 23190 Tyi4 a y40s a sco ae 36.26. x: 19°) yagnee 
BON Daeleteres 42050 ; SESGi4 TOO cease pears 62.10 < 10°) SaGsee 
AGie eaten ee Wes 73200 6 26.4 SL EO ssoratc wise 100.60 10° Veyaaae 
EO yi sececs eel sea eeOe O62 “| "206% wecsat 156. X 10° “TrGsee 


TABLE XIII. 


CRITICAL TEMPERATURES (7) AND PRESSURES IN ATMOSPHERES (P), AT 
THEIR CRITICAL TEMPERATURES, OF VARIOUS GASES. 


Fue yA a3 Da 
Hydrogén.nisn 5. us — 174. 99. | Carbon dioxide... 30.9 aye Ne 
Nitrogen... 4sse04 «1245 42. | Sulphur dioxide... 155.4 79< 
COZV POU est ae ee OnG 49. 


TABLE XIV. 


COEFFICIENTS OF CONDUCTIVITY FOR HEAT (X) IN C. G.S. UNITs, Iv 
WHICH Q IS GIVEN IN LESSER CALORIES. 


YASS. sins eve Seeder naee ee ne 0.30 MECCUEY fire aine os a) bed naan om 
COPpetsn «cscs 06 6.6 leeee web ab ae teik Paraffin..... os es 0 0 39 Senne 0.00014 
Gla6S piace Se abe nts. caeia’s OFS 0.0005) Silver”... 4.5 selcfaeaname isu s tees 
Tees cients aeen ale era nea alate tees 0.0057 | Vulcanized india-rubber..... 0.00009, 
Tron. ie een wees we higa © ute 0.16 Waters ai ctas wen PE eee er 


TABLES, 


TABLE. XV. 


503 


ENERGY PRODUCED BY COMBINATION OF I GRAM OF CERTAIN SUBSTANCES 


WITH OXYGEN. 


Gram-degree of Heat. 


Meeroaie 10tmine CO... .ceics © 2141 

a‘ sf CO, ele Bly wibvie. 6 0 4.0 8000 
Carbon monoxide, forming CO... 2420 
MCAS) WG Bees tices e's yc 602 
Brpcroven, HO... eee ees 34000 
Marsh gas, CO, and H,0........ 13100 
OO, oss ww a-snnie s Per reek 1301 

‘TABLE XVI. 


Energy in ergs. 
8.98 X 10! 


30) x ‘10! 
Mey Die Ga Oot 
Rac IOS 
PV AS 
«50° x r0!"! 
46 X 10! 


nord NO Bw 


ATOMIC WEIGHTS AND COMBINING NUMBERS, 


Atomic Weight. 


Combining Number, 


SEPMEMEEMRU SLT etait pine an'n ss soe a= pve Sd 27.04 g.O1 
SRRIITOUEEES 5 <\es).2'nti-dvie Siwleie ao shale wule & 63.18 (cupric) Z1250 
ee Gtreras » aieiiase ie ek shear etm ier cle toa bs (cuprous) 63.18 
SOMMERS iS sinus cecad «c,4d.cau ae sis 196.2 65.4 
ERTL Rtas a w\Od'e wath, av soa pares 1 rt 
UC ae sahtete'is e's wlavelete ou 55.88 (ferric) 18.63 
TS Ba oe ee Se s (ferrous) 27.94 
Sy I PR Ae Aa SreLOQ.e,. (mercuric): 99.9 
ET Shas ohnts ais ss 0 o'ssu(a oe alas es (mercurous) I99.8 
REGIE apy wis <'c's ices A a's 25.00 9's ee 58.6 29.3 
ROI AY wy 'x G's od a's eis'einon wold ea dace 15.96 7.98 
ROPER ies ay 6 an 00a a a,4.¥ 000,04 6 415 194.3 64.8 
INNO cucis os x'ad oc le% Weve s 6% TO7s5 107.7 
DCM era tis vesys Ceced'e ce oe eens 64.88 32.44 


TABLE XVII. 


MOLECULAR WEIGHTS AND DENSITIES OF GASES. 


Simple Gases. 
Atomic Weight. 


Spier. 77 = %, 


Mass in 1 litre. 


MTG la, og cee ccens occas 70.75 a5 a% 3.167 
Co re 2.00 I.00 0.0895 
Nitrogen, Ne..... FIONN ie : 28.024 14.012 1.254 
ST MCLs sie o's con ccvses 31.927 15.96 1.429 


504 ELEMENTARY PHYSICS, 


Compound Gases. 


Atomic Weight. SP. Gi .342 ee Mass in litre, — 
Garbonicioxide sO... acts «tee 27.937 14.97 1.22 
Carvonie dioxide,.COs. 4.08 sss 43.90 21.95 1.965 
Hydrochloric acid, HCI... .5. 30376 18.188 1.628 


Vapor of water, H2O.......... 17.96 8.98 0.804 
PULMOSPHEMMCA Aly cusses e's 0 sis 1.264 


TABLE XVIII. 
ELECTROMOTIVE FORCE OF VOLTAIC CELLS. ' 
daaniell cies eis 1.1 volt. | Grove........ 1.88 volt. | Clark... 1.435 voltages bes 


Electromotive force of Clark cell for any temperature ¢ is 
1.435[I — 0.00077(¢ — 15)]. 


TABLE XIX. 


ELECTRO-CHEMICAL EQUIVALENTS. 
Crams per second deposited by the electromagnetic unit current, 


Hydrogen, 0.0001038. 


To find the electro-chemical equivalents of other substances, multiply the | : j 
electro-chemical equivalent of hydrogen by the combining number of the sub- 
stance. 


TABLE 2X3 
ELECTRICAL RESISTANCE. 
Absolute resistance # in C. G. S. units of a centimetre cube of the substance. — fe 
Temperature coefficient, a. Rr = RoI + ae), | 


Ne a. 
Aluminigogies.ouces van eC ar arres - 2889 Pon 
Copper 25 ais evelele wine, eaoeheaal » sie 1611 0.00388 
German SUY@R Sn) 25 <tes onsean'e wlan 20763 0.00044 — 
Golde wise Os areaa ee eh be ayers "< 2041 0.00365 
lrone-n wee eee Leake eee 9638 oat ae 
MGtEUlY 6.1.5. ases wels ates meee ‘ 94340 0.00072 
Platinuni... 3k tee eat cae ey re! 89$2 0.00376 
Platinum silver,.2 Pt. 1 Ags:.cn.. 24190 0.00031 
SilVers dork eae muoks sa sete ois wl eee 1580 0.00377 
Zinc......- Pena eae a otse oh caeheee ens 5581 0.00365 


Carbon (Carré’s electric light) 
DRE EON RE a gy 5.0544 yr a 4 5.6 ace eel sd aaa eo 6 ns 
PURO MU MIAMEAC OA veces sesh a shcsece ae venet 


66 3 66 ° 
oO 


RNC ON eg oy 5d g's «Waseca buco SiSloaie Soni wlecese 
ete Moe eal .8. che Warn Uae aie, scandent 
oy SS 1 Hs 0 ee eran 


Copper, sulphate + 45 H.O 


TABLE {AXT: 


TABLES. 


ove eorer eee eee ees ee een ere 


eee eee eee eee ee eeneere 


see eee ec eee eee ee eee eee ee tenes 


HSH HesTmN OW WN W 


INDICES OF REFRACTION. 


Kind of 
Light. 


Index. 
Soft crown glass..... 1.5090 
1.5180 
1.5266 
Dense flint glass..... 1.6157 
1.6289 
1.6453 
PERU SENG yh sasesc.e. cme. 1.5300 
1.5490 
1.5613 
We ee hee 2.47 


Hil ek 1.532 
Ordinary Index. 


Iceland spar..... 
MUURULI os wean dass 


Fraunhofer’s line A (edge), 7593. 
6867. 


DOOM ramnoamey 


1.658 
1.544 


Canada balsam...... ry 
WH ALOT rs Coke ae cee os 


Carbon disulphide.... 


Air at 0°, 760 mm.... 


Kind of Light. 
D 


SS oS SH SS SH SS SS aS 


D 


TABLE XXII. 


WAVE LENGTHS OF LIGHT—ROWLAND’S DETERMINATIONS. 


B 


eé 


ce 


6562 


4861 


975 tenth metres. 
382 


.965 
5896. 
5890. 
5270. 
5183. 
.428 
4307. 


080 
125 
429 
rit 


g6I 


Ae TON 
As she id Co pe 
.87X 1074 


23% 10°° 
Soto e SON 


528 Red 
B 


raat 
. 330 

344 H 
614 A 
.646 E 
.684 G 
.00029 A 
. 000296 - 
.000300 H 


Extraordinary Index. 


1.486 
1.553 


Nuss ° 
» rit Y 
he \ , 
By a 
506 ELEMENTARY PHYSICS. | oo 


TABLE XXIII. 


ROTATION OF PLANE OF POLARIZATION BY A QUARTZ PLATE, I MM. THICK, 
2). 
CUT PERPENDICULAR TO AXIS, co - 


| 


ALAR aat Ee had 12°, 668°) JB) 480.) a vole oe 
B a) 8 0,610 @ Lm eececerseeee es « 15°.746 F avec efeeéten eeu s cleus w eoeeee 32° 773i 
CAs (2 oe MURS Saw eae ee 7°egteus (Gay ope akes PD 2 3. 42°.604 — 


Dass cies eens eeeecece Ce 21° .727 H ee eeeerees eeeeereeaeeseees 51°.193 | 


TABLE XXIV. 


VELOCITIES OF LIGHT. 


Cm. per Sec. Cm. per Sec. 
Michelson, 1879...... 2.99910 X 10! | Foucault, 1862........ 2.98000 X 10? 
Michelson, 1882...... 2.99853 X 10 | Cornu, 1874.......4+. 2.98500 Gan 
Newcomb, 1882...... 2.99860 X 10! -| Cornu, 1878.......++. 2.09000). unmum < 


s : p , Y 
THE RATIO BETWEEN THE ELECTROSTATIC AND ELECTROMAGNETIC UNITs. yi 


per Sec. ae per Sec, a 
Weber and Kohlrausch “i ee < To SoirBisner ee ee 1eeee 2.920 0 10mm 


Wi LHOMmSOR chs «owes 2.825 X10 | Klemenéié. bier ane 3.018 X 10% 
Maxwells vc.d.....00s) 2288 © Dé Tolls A imstedtenecsmeameec ee . 3.007 107M x 
Ayrton and Perry....: 2.98 rol! | (Colleyicdi 00 ee ceus «cen 10> 

Daler LAOMSOT snes 4105 mi Oe a 


TENG Ox 


ABERRATION, spherical, 426; chromatic, 455 

Aberration of fixed stars, 432 

Absolute temperature, zero of, I91; scale of, 212 

Absorption, 103; coefficient of, 104; of gases, 104; of radiant energy, 463; of 
radiations, 466; by gases, 467; relation of, to emission, 470 

Acceleration, 14; angular, 49 

Accelerations, composition and resolutions of, 16 

Achromatism, 455 

Acoustics, 353 

Adhesion, 87 

Adiabatic line, 194 

Aggregation, states of, 85 

Air-pump, 137; réceiver of, 137; plate of, 138; theory of Sprengel, 134; Spren- 
gel, 139; Morren, 140 

Airy, determination of Earth’s density, 80 

Alloys, melting points of, 176 

Ampére, relation of current and magnet, 273; mutual action of currents, 311; 
equivalence of circuit and magnetic shell, 314; theory of magnetism, 315 

Ampere, a unit of electrical current, 309 

Amplitude of a simple harmonic motion, 18; of a wave, 23; its relation to in- 
tensity of light, 438 

Analyzer, 481 

Andrews, critical temperature, 184; heat of chemical combination, 2o1, 

Aneroid, I41 

Angles, measurement of, g; unit of, 9 

Animal heat and work, 218 

Antinode, 362 

Aperture of spherical mirrors, 411 

Apertures, diffraction effects at, 443 

Archimedes, his principle in hydrostatics, 124 

Aristotle, his theory of vision, 396 


508 INDEX. 


Astatic system of magnetic needles, 317 

Atmosphere, homogeneous, 115; pressure of, 123; how stated, 124 

Atoms, 85 

Attraction, mass or universal, 67; constant of, 80 

Avenarius, experiments in thermo-electricity, 342; thermo-electric formula, 345 

Axis of rotation, 52; of shear, 134; of floating body, 125; magnetic, 224, 228; 
of spherical mirror, 411; optic, of crystal, 474, 489 


BALANCE, 76; hydrostatic, 125 

Barometer, 122; Torricellian form of, 123; modifications of, 124; preparation 
of, 124 

Beam of light, 422. 

Beats of two tones, 385; Helmholtz’s theory of, 385; KGnig’s theory of, 385; 
Cross’s experiment on, 387 

Beetz, his experiment on a limit of magnetization, 245 

Berthelot, heat of chemical combination, 202 

Berzelius, his electro-chemical series, 287 

Bidwell, view of Hall effect, 316 

Bifilar suspension, 268, 321 

Biot, law of action between magnet and electrical current, 298 

Biot and Savart, action between magnet and electrical current, 297 

Bodies, composition of, 85; forces determining structure of, 87; isotropic and 
eolotropic, 108 

Boiling. See Ebullition, 182 

Boiling point, 182 - 

Bolometer, depends upon change of resistance with temperature, 279; used to 
study spectrum, 459 

Boltzmann, specific inductive capacity of gases, 264 

Borda, his pendulum, 74; his method of double weighing, 78 

Bosscha, capillary phenomena in gases, 96 

Boutigny, spheroidal state, 183 

Boyle, his law for gases, 110; limitations of, 141; departures from, 185 

Bradley, determined velocity of light, 433 ~ 

Breaking weight, I19 

Brewster, his law of polarization by reflection, 480 


CAGNIARD-LATOUR, critical temperature, 183 

Calorie, 151; lesser, 151 

Calorimeter, Black’s ice, 153; Bunsen’s ice, 153; water, 155; thermocalorime- 
ter of Regnault, 157; water equivalent of, 156 

Calorimetry, 153; method of fusion, 153; of mixtures, 155; of comparison, 
156; of cooling, 157 


, 


i ee i i 


/ 
q 
‘ 
7 
. 
j 


INDEX. 509) 


Camera obscura, 427 

Capacity, electrical, 255; unit of, 256; of spherical condenser, 258; of plate 
condenser, 260; of freely electrified sphere, 260; of Leyden jar, 261 

Capacity, specific inductive, 257; relation of, to index of refraction, 264, 495; 
relation of, to crystallographic axes, 264 

Capillarity, facts of, 89; law of force treated in, 90; equation of, 94; in gases,. 
96; Plateau’s experiments in, 97 

Carlini, determination of Earth’s density, 80 

Carlisle, his apparatus for electrolysis of water, 283 

Carnot, his engine, 206; his cycle, 207 

Cathetometer, 6 

Cavendish, experiment to prove mass attraction, 69; determination of Earth’s 
density, 79; determined force in electrified conductor, 249; discovered. 
specific inductive capacity, 257 

Caustic curve, 426; surface, 426 

Central forces, propositions connected with, 60 

Centrobaric bodies, 45 

Charge, unit, electrical, 252; energy of electrical, 262 

Chemical affinity measured in terms of electromotive force, 286 

Chemical combination, heat equivalent of, 201 

Chemical separation, energy required for, 218; gives rise to electromotive 
force, 285 

Chladni’s figures, 377 

Christiansen, anomalous dispersion in fuchsin, 472 

Circle divided, 9 

Circuit, electrical, equivalence of, to magnetic shell, 300, 305; direction of lines. 
of force due to, 310 

Clark, his standard cell, 293; its electromotive force, 293; his potentiometer, 334. 

Clarke, his atomic weights used, 176 

Clausius, his principle in thermodynamics, 206; his theory of electrolysis, 289. 

Clement and Desormes, determination of ratio of specific heats of gases, I95 

Coercive force, 223 

Cohesion, 87 

Collimating lens, 451, 459. 

Collision of bodies, 29 

Colloids, 86; diffusion of, 107 

Colors of bodies, 467; produced by a thin plate of doubly refracting crystal in, 
polarized light, 485; by a thick plate, 486 

Colors and figures produced by a thin plate of doubly refracting crystal in 
polarized light, 489, 490, 491 

Comparator, 8 

Compressibility, 84 


510 INDEX. 


Compressing pump, 140 

Compressions, 109 

Concord, musical, 367 

Condenser, electrical, 256; spherical, 258; plane, 260 

Conduction of electricity, 246 

Conductivity for heat, 164; measurement of, 165; changes of, with temperature, 
167; of crystals, 167; of non-homogeneous solids, 167; of liquids, 167 

Conductivity, specific electrical, 278 

Conductors, good, 247; poor, 247 

Contact, angles of, 95 

Continuity, condition of, 128; for a liquid, 128 

Convection of heat, 161 

Copernicus, his heliocentric theory, 67 

Cords, longitudinal vibrations of, 374; transverse vibrations of, 375 

Cornu and Baille, determination of Earth’s density, 80 

Coulomb, his laws of torsion, 116; his torsion balance, 116; law of magnetic 
force, 225; distribution of magnetism, 228; law of electrical force, 249 

Coulomb, a unit of. quantity of electricity, 252 

Counter electromotive force, 279; general law of, 279; of decomposition, 
measure of, 285; of polarization, 291; of electric arc, 348 

Couple, 44; moment of, 44 

Critical angle of substance, 408 

Critical temperature, 183 

Crookes, invented the radiometer, 192; his tubes, 351; explanation of phe- 
nomena in tubes, 352 

Cross, experiment on beats, 387 

Crystal systems, 86 

Crystalloids, 86; diffusion of, 107 

Crystals, conductivity of, for heat, 167; specific inductive capacity of, 2643 elec- 
trification of, by heat, 264; optic axis of, 474; principal plane of, 475; vary- 
ing elasticity in, 478; varying velocity of light in, 479; effects of plates of, 
on polarized light, 483, 486, 489, 490, 491; uniaxial, 489; biaxial, 489; 
optic axes of biaxial, 489 

Ctesibius, invented force-pump, 122 

Cumming, reversal of thermo-electric currents, 342 

Current, electrical, 275; effects of, 272; electrostatic unit of, 275; strength, 275; 
strength depends on nature of circuit, 276; set up by movement of a liquid 
surface, 296; electromagnetic unit of, 307; practical unit of, 309; direction 
of lines of force due to, 298, 310; mutual action of two, 310; Ampére’s law 
for the mutual action of, 311; deflected in a conductor by a magnet, 315; 
measured in absolute units, 321; Kirchhoff’s laws of, 331 

Current, extra, 325 ° 


INDEX. 511 


Current, induced electrical, 321; quantity and strength of, 323; measured in 
terms of lines of force, 323; discovered by Faraday, 324; Lenz’s law of, 
325; Faraday’s experiments relating to, 325 

Cycle, Carnot’s, 207; illustrated in hot-air engines, 216 


DALTON, his law of vapor pressure, 181 

Daniell’s cell, 291; electromotive force of, calculated, 292 

Dark lines in solar spectrum, explanation of, 469 

Davy, his melting of ice by friction, 145; his electrolysis of caustic potash, 283 

Declination, magnetic, 233 

Density, II 

Density, magnetic, 227 

Depretz and Dulong, measurement of animal heat by, 219 

Dew point, 203; determination of, 203 

Dialysis, 107 

Diamagnet, distinguished from paramagnet, 239, 242 

Diamagnetism, 237; explanation of, by Faraday, 237; Ry Thomson, 238; on 
Ampeére’s theory, 315; by Weber, 315 

Diaphragm, vibrations of, 378 

Dielectric, 256; strain in, 263 

Diffraction of light, 443; at narrow apertures, 443; at narrow screens, 445; 
grating, 446; phenomena due to, 452 

Diffusion, 103; of liquids, 104; coefficient of, 105; through porous bodies, 106; 
through membranes, 106; of gases, 107 

Dilatability, 84 

Dilatations, 109 

Dimensional equation, 10 

Dimensions of units, 10 

Dip, magnetic. 233 

Discord, in music, 367 

Dispersive power of substance, 454 

Dispersion, normal, 409, 453; anomalous, 472 

Dissociation, 201; heat equivalent of, 201 

Distribution of electricity on conductors, 251 

Dividing engine, 7 

Divisibility, 84 

Double refraction, in Iceland spar, 474; explanation of, 475; by isotropic sub- 
stances when strained, 490 

Draper, study of spectrum in relation to temperature, 471 

Drops, in capillary tubes, 101; Jamin’s experiments on, 102 

Dulong and Petit, law connecting specific heat and atomic weight, 176; formula 
for loss of heat by radiation, 470 


512 INDEX. 


Dutrochet, his definition of osmosis, 106 
Dynamics, II 

Dynamo-machine, 328 

Dyne, 26 


Ear, tympanum of, 379 

Earth, density of, 79 

Ebullition, 180, ona of, 182; causes affecting, 182 

Edlund: study cf counter electromotive force of electric arc, 348 

Efflux through narrow tubes, 88; of a liquid, 129; quantity of, 134 

Elasticity, 84, 108; modulus and coefficient of, 110; voluminal, of gases, IIo; 
of liquids, 112; of solids, 113; perfect, 118; of tension, 114; of torsion, 
115; of flexure, 118; limits of, 118 

Elasticity of gases, 110, 193; at constant temperature, I93; when no heat enters 
or escapes, 193; ratio of these, 198; determined from velocity of sound, 
199 

Electric arc, 348; counter electromotive force of, 348 

Electric discharge, in air, 348; in rarefied gases, 350 

Electric pressure, 255 

Electrical convection of heat, 347 

Electrical endosmose, 288; shadow, 349 

Electrical machine, 268; frictional, 268; induction, 269 

Electricity, unit quantity of, 252; flow of, 253, 274 

Electrification by friction, 246; positive and negative, 246, 248 

Electro-chemical equivalent, 284 

Electrode, 282 

Electrodynamometer, 321 

Electrolysis, 282; bodies capable of, 282; typical cases of, 283; influenced by 
secondary chemical reactions, 283; Faraday’s laws of, 284; theory of, a 
modified by outstanding facts, 288; Clausius’ view of, 289 

Electrolyte, 282 

Electromagnet, 314 

Electromagnetic system of electrical units, basis of, 307 

Electrometer, 265; absolute, 265; method of use of, 267; quadrant, 267; capil- 
lary, 294 

Electromotive force, 274; measured by difference of potential, 274; means of 
setting up, 280; measured in heat units, 286; a measure of chemical affinity, 
286; of polarization, 291; theories of, of voltaic cell, 293; due to mo- 
tion in magnetic field, 321; measured in terms of lines of force, 323; de- 
pends on rate of motion, 323; electromagnetic unit of, 326; practical unit 
of, 326; at a heated junction, 341; required to force spark through air, 349 

Electromotive force, counter. See Counter electromotive force, 279, etc. 


INDEX. 513 


Electromotive forces, compared by Clark’s potentiometer, 334 

Electrophorus, 269 

Electroscope, 265 

Electrostatic system of electrical units, basis of, 252 

Elements, chemical, 85; electro-positive and electro-negative, 286 

Emission of radiant energy, 468; relation of, to absorption, 470 

Endosmometer of Dutrochet, 107 

Endosmose, 106 

Endosmose, electrical, 288 

Energy, 31; potential and kinetic, 31; and work, equivalence of, 32; unit of, 32; 
conservation of, 32; of fusion, 179; of vaporization, 200; sources of terres- 
trial, 217; of sun, 221; dissipation of, 221 

Engine, efficiency of heat, 205; reversible, 206; Carnot, 206; efficiency of re- 
versible, 206, 209, 210, 211; steam, 214; hot-air, 215; gas, 215; Stirling, 

Pe2rOy  hider, 216 

Eolotropic bodies, 108 

Epoch of a simple harmonic motion, 21 

Equatorial plane of a magnet, 224 

Equilibrium, 29 

Equipotential surface, 37 

Erg, 32 

Ether, 85; luminiferous, 397; velocity of, in moving body, 435; interacts with 
molecules of bodies, 473; transmits electrical and magnetic disturbances, 
495 

Ettinghausen, view of Hall effect, 316 

Evaporation, 180; process of, 180 

Exosmose, I06 

Expansion, of solids by heat, 168; linear, 168; voluminal, 168, 173; coefficient of, 
168; factor of, 169; measurement of coefficient of, 169, 173; of liquids by 
heat, 170; absolute, 170, 174; apparent, 170; of mercury, absolute, 170; 
apparent, 171; of water, 174; of gases by heat, 185; coefficient of, 186; 
heat absorbed and work done during, 194 

Extraordinary ray,-475; index, 475 

Eye, 427; estimation of size and distance by, 428 

Eye-lens or eye-piece, 431; negative or Huyghens, 456; positive or Ramsden, 
457 


FARAD, a unit of electrical capacity, 256 
Faraday, discovery of magnetic induction in all bodies, 237; explanation of 
this, 237; experiment in electrical induction, 247; on force in electrified 
body, 249; theory of electrification, 256; theory illustrated, 263; explana- 
tion of residual charge, 264; showed that discharge of jar can produce ef- 
33 


514 INDEX. 


fects of current, 274; nomenclature of electrolysis, 282; voltameter, 285; 
division of ions, 286; theory of electrolysis, 287; chemical theory of electro- 
motive force, 293; electromagnetic rotations, 306; induced currents, 324; 
effect of medium on luminous discharge, 350; electromagnetic rotation ot 
plane of polarization, 493 

Favre and Silbermann, studied heat of chemical combination, 202; verified con- 
nection of electromotive force and heat units, 286; value of heat equivalent, 
292 

Ferromagnet. See Paramagnet, 237 

Field of force, 27; strength of, 27 

Filament, in a fluid, 129 

Films, studied by Plateau, 98; interference of light from, 441 

Fizeau, introduced condenser in connection with induction coil, 329; deter- 
mined velocity of light, 433; velocity of light in a moving medium, 435 

Flexure, elasticity of, 118 

Floating bodies, 125 

Flow of heat, 162; across a wall, 162; proportional to rate of fall of tempera- 
ture, 163; alonga bar, 165 

Fluid, body immersed in a, 125; body floating on a, 125 

Fluids, distinction between solids and, 119; mobile, viscous, 119; perfect, 120 

Fluids, motions of. Sze Motions of a fluid, 128 

Fluorescence, 472 

Focal line, 425 

Focus, of spherical mirror, 413; real, conjugate, 413; principal, 414; virtual, 414 

Force, 26; unit of, 26; field of, line of, tube of, 27; defined by potential, 34, 
within spherical shell, 38; outside sphere, 41; just outside a spherical shell, 
42; just outside a flat disk, 42; moment of, 43 

Force, capillary, law of, 90 ; 

Force, electrical, in charged conductor, 249; law of, 249; screen from, 253; 
just outside an electrified conductor, 255 

Force, magnetic, law of, 224; due to bar magnet, 230; within a magnet, 238; 
between magnet and current element, 297, 298; between magnet and long 
straight current, 299; due to magnetic shell, 304 

_ Forces, composition and resolution of, 29; resultant of parallel, 43; central, 60 

Forces, determining structure of bodies, 87; molecular, 87, 108; of cohesion, 
87; of adhesion, 87 

Foucault, his pendulum, 52; determined velocity of light, 434; his prism, 482 

Fourier, hisetheorem, 25 

Franklin, complete discharge of electrified body, 248; experiment with Leyden 
jar, 263; identity of lightning and electrical discharge, 350 

Fraunhofer, lines in solar spectrum, 458 

Freezing point, change of, with pressure, etc., 177 


INDEX. 515 


Fresnel, interference of light from two similar sources, 439; his rhomb, 489; 
explanation of rotation of plane of polarization by quartz, 491 

Friction, laws of, 88; coefficient of, 88; theory of, 89 

Fusion, 176; heat equivalent of, 178; energy necessary for, 179; determination 
of heat equivalent of, 179 


GALILEO, the heliocentric theory, 68; measurement of gravity, 70; path of pro- 
jectiles, 81; weight of atmosphere, 123 

Galvani, discovered physiological effects of electrical current, 272 

Galvanometer, 316; Schweigger’s multiplier, 316; sine, 317; tangent, 318 

Gas, definition of, 180 

Gases, 85; absorption of, 104; diffusion of, 107; elasticity of, 110; liquefaction 
of, by pressure, 142, 184; departure of, from Boyle’s law, 185; coefficient 
of expansion of, 186; pressure of saturated, 186 

Gases, kinetic theory of. See Kinetic theory of gases, 188 

Gauss, theory of capillarity, 91 

Gay-Lussac, law of expansion of gases by heat, 185 

Geissler tubes, 351 

Gilbert, showed Earth to be a magnet, 233 

Graham, his osmometer, 107; method of dialysis, 107 

Grating, diffraction, 446; element of, 447; pure spectrum produced by, 447; nor- 
mal and oblique incidence, 447; with irregular openings, 450; wave lengths 
measured by, 450; Rowland’s curved, 452 

Gravitation, attraction of, 67 

Gravity, centre of, 45 

Gravity, measurement of, 69; value of, 70 

Grotthus, theory of electrolysis, 287 

Grove, his gas battery, 291 

Grove’s cell, 292 

Gyration, radius of, 56 

Gyroscope, 53 


HALL, deflection of a current in a conductor, 315 

Halley, theory of gravitation, 68 

Hamilton, prediction of conical refraction, 490 

Harmonic tones of pipe, 373 

Harris, absolute electrometer, 265 

Heat, effects of, 143; production of, 144; nature of, 144; a form of energy, 145; 
unit of, 151; mechanical unit of, 151; mechanical equivalent of, 158; Joule’s 
determination of, 158; Rowland’s, 159; transfer of, 161; convection of, 
161; internal, of Earth, a source of energy, 221; developed by the electri- 
eal current, 273, 275; generated by absorption of radiant energy, 463 


516 INDEX. 


Heat, conduction of. See Flow of, 162 

Heat, atomic, 175 

Helmholtz, vortices, 135; theory of solar energy, 221; law of counter electro- 
motive force, 280; theory of capillary electrometer, 295; resonators, 352;. 
vowel sounds, 383; theory of beats, 385; interaction of ether and molecules. 
of bodies, 473 

Herschel, study of spectrum, 460 

Hirn, work done by animals, 219 

Holtz, electrical machine, 270 

Hooke, theory of gravitation, 68 

Hopkinson, relation between index of refraction and specific inductive ca~ 
pacity 496 

Horizontal intensity of Earth’s magnetism, 233; measurement of, by standard 
magnet, 234; absolute, 235 

Humidity, absolute, 202; relative, 204 

Huyghens, theorems of, on motion in a circle, 68; views of, respecting gravita- 
tion, 68; principle of wave propagation, 356 

Hydrometer, 127 

Hydrostatic balance, 125 

Hydrostatic press, 121 

Hygrometer, Alluard’s, 203 

Hygrometry, 202 


Ick, density, of, 177; melting point of, used as standard, 176 

Iceland spar, 474; wave surface in, 476 

Images, formed by small apertures, 402; virtual, 409; by successive reflection, 
410; by mirrors, 419; by lenses, 421; geometrical construction of, 422 

Impenetrability, 4 

Impulse, 26 

Incidence, angle of, 406 

Inclined plane, 47 

Induced magnetization, coefficient of, 239 

Induction coil, 328; condenser connected with, 329 

Induction, electrical, 247 

Induction, magnetic, 223, 237; definition of, 239 

Induction ot currents, 321 

Inertia, 4, 30, centre of, 44; moment of, 56 

Insulator, electrical, 247 

Interference of light, cause of propagation in straight lines, 397; from two simi= 
lar sources, 436, experimental realization of, 439; from thin films, 441 

Internode, 362 

Intervals, 36% 


INDEX. aha, 


Tons, 282; electro-positive and electro-negative, 286; arrangement of, by Fara- 
day, 286; by Berzelius, 287; wandering of the, 288 

Isothermal line, 193 

Isotropic bodies, 108 


JAMIN, drops in capillary tubes, 102 

Jolly, determination of Earth’s density, 80 

Joule, equivalence of heat and energy, 145, 205; mechanical equivalent of heat, 
158; expansion of gas without work, 188; limit of magnetization, 245; law 
of heat developed by electrical current, 279; electromotive force in heat 
units, 286; development of heat in electrolysis, 288 

Jurin, law of capillary action, 99 


KALEIDOSCOPE, 410 

Kater, his pendulum, 75 

Kepler, laws of planetary motion, 67 

Kerr, optical effect of strain in dielectric, 263; rotation of plane of polarization 
by reflection from magnet, 495 

Ketteler, interaction of ether and molecules of bodies, 473 

Kinematics, II 

Kinetics, 11 

Kinetic theory of gases, 188 

Kirchhoff, laws of electrical currents, 331; spectrum analysis, 460 

Kohlrausch, value of electro-chemicalt equivalent, 292 

KGnig, A., modification of surface tension by electrical currents, 295 

KG6nig, R., manometric capsule, 353; pitch of tuning-forks made by, 370; 
boxes of his tuning-forks, 378; quality as dependent on change of phase, 
381; investigation of.beats, 385 

Kundt, experiment to measure velocity of sound, 394; anomalous dispersion, 
472 


LANG, counter electromotive force of electric arc, 348 

Langley, his bolometer, 279; wave lengths in lunar radiations, 452 

Laplace, theory of capillarity, 91 

Lavoisier, measurement of animal heat, 218 

Least time, principle of, 4or1 

Length, unit of, 4; measurements of, 5 

Lenses, 417; formula for, 417; forms of, 418; focal length of, 418; images 
formed by, 421; optical centre of, 421; thick, 423; of large aperture, 423; 
aplanatic combinations of, 427; achromatic combinations of, 455 

Lenz,-law of induced currents, 325 


eo ied 


518 INDEX. 


Le Roux, experiments in thermo-electricity, 342; electrical convection of heat 
in lead, 347 

Lever, 46 

Leyden jar, capacity of, 261; dissected, 263; volume changes in, 263; residual 
charge of, 264 

Light, agent of vision, 396; theories of, 396; propagated in straight lines, 397; 
principle of least time, gor; reflection of, 404; refraction of, 405; ray of, 
beam of, pencil of, 422 

Light, velocity of, determined from eclipses of Jupiter’s satellites, 432; from 
aberration of fixed stars, 432; by Fizeau, 433; by Foucault, 434; by Michel- 
son, 434; in moving medium, 435 

Light, electromagnetic theory of, 495 

Lightning, an electrical discharge, 350 

Lines of magnetic force, positive direction of, 304; measure of strength of field 
in, 309; relation of, to moving magnetic shell or current, 309 

Lippmann, electrical effects on capillary surface, 294; capillary electrometer, 
296; production of current by modification of capillary surface, 296 

Liquefaction, 184; of gases, by pressure, 184 

Liquids, 85; modulus of elasticity of, 112 

Lissajous, optical method of compounding vibrations, 384 

Loudness of sound, 365 


MACHINE, 48; efficiency of, 48; electrical, 268; dynamo- and magneto-, 328 

Magnet, natural, 223; bar, relations of, 228 

Magnetic elements of Earth, 233 

Magnetic force. See Force, magnetic, 224 

Magnetic inductive capacity, 239 

Magnetic shell, 231; strength of, 231; potential due to, 232; equivalence of, to 
closed current, 300 

Magnetic system of units, basis of, 226 

Magnetism, fundamental facts of, 223; distribution of, in magnet, 227; deter- 
mination of, 228; theories of, 244; Ampére’s theory of, 315 

Magnetization, intensity of, 226 

Magneto-machine, 328 

Magnifying glass, 430 

Magnifying power, 429 

Manometer, 140 

Manometric capsule, 353 

Mariotte, study of expansion of gases, IIO 

Maskelyne, determination of Earth’s density, 79 

Mass, I1; unit of, 9 

Masses, comparison of, 9 


INDEX. 519 


Matter, I; constitution of, 84 

Matthiessen, expansion of water, 174 

Mayer, views concerning work done by animals, 219 

Maxwell, proposed unit of time, 83; coefficient of viscosity of a gas, 89; defini- 
tion of magnetic induction, 239; theory of electrification, 256; explanation 
of residual charge, 264; relation between specific inductive capacity and 
index of refraction, 264, 495; suggested test of Weber’s theory of diamag- 
netism, 315; measurement of v, 337; force on magnet due to moving elec- 
trical charge, 338; electromagnetic theory of light, 397, 495 

Mechanical powers, 46 

Melloni, use of thermopile, 341 

Melting point of ice, 176; of alloys, 176; change of, with pressure, 177 

Mercury, expansion of, by heat, 170, I7I 

Metacentre, 125 

Michelson, determined velocity of light, 434 

Michelson and Morley, velocity of light in moving medium, 435 

Micrometer screw, 6 

Microscope, simple, 430; compound, 430 

Mirrors, plane, 409; spherical, 410; images formed by, 419; of large aperture, 
423; not spherical, 424 

Modulus of elasticity. See Elasticity, 110; Young’s, 115 

Molecular action, radius of, 90 

Molecule, 84; structure of, 87; kinetic energy of, proportional to temperature, 
I9gI; mean velocity of, 103 

Moment, of force,.43; of momentum, 43; of couple, 44 

Moment of inertia, 56; of rod, 57; of plate, 58; of parallelopiped, 59; experi- 
mentally determined, 60 

Moment of torsion, 116; determination of, 117 

Moment, magnetic, 226; changes in, 243; depends on temperature, 244; on 
mechanical disturbance, 244 

Momentum, 14; conservation of, 29; moment of, 43 

Motion, 12; absolute angular, 12; simple harmonic, 18; Newton’s laws of, 27; 
in acircle, 61; in an ellipse, 64 

Motions, composition and resolution of, 16; of simple harmonic, 22; of a fluid, 
128; optical method of compounding, 384 

Miiller, J., limit of magnetization, 245 


NEWTON, laws of motion, 27; central forces, 61; law of mass attraction, 68; 
quantity of liquid flowing through orifice, 134; theory of light, 396; inter- 
ference of light from films, 442; composition of white light, 453; chromatic 
aberration, 455; law of cooling, 470 

Nichols, study of radiations, 471 


520 INDEX. 


Nicholson and Carlisle, decomposition of water by electrical current, 273 
Nicol, prism, 482 

Node, 362 

Noise, 365 


OBJECTIVE, 431 

Ocean currents, energy of, 218 

Oersted, his piezometer, 112; relation between magnetism and electricity, 273 

Ohm, law of electrical current, 276, 277 

Ohm, a unit of electrical resistance, 330; various values of, 330; determination . 
of, 330 

Optic angle, 429; axis of crystal, 474, 489 

Optics, 396 o ‘ 

Ordinary ray, 475; index, 475 

Organ pipe, 371; fundamental of, 373; harmonics of, 373; mouthpiece of, 373; 
reeds used with, 373 : 

Osmometer, Graham’s, 107 

Osmosis, 106 

Overtones, of pipe, 373 


PARALLELOGRAM, of motions, etc., 16; of forces, 29 

Particle; 12 

Pascal, pressure in liquid, 120; pressure modified by gravity, 121; barometer, 
123 

Path,:12 

Peltier, heating of junctions by passage of electrical current, 273; effect, 274, 340 

Pencil of light, 422 

Pendulum, Foucault’s, 52; simple, 70; formula for, 71; physical, 72; Borda’s, 
74; Kater’s, 75 

Penumbra, 402 

Period, of a simple harmonic motion, 18; of a wave, 23 

Permeability, magnetic, 239 

Pfeffer, study of osmosis, 107 

Phase, of a simple harmonic motion, Ig 

Phonograph, 378 

' Phosphorescence, 472 

Photometer, Rumford’s, Foucault’s, Bunsen’s, 465 

Photometry, 464 

Piezometer, Oersted’s, 112; Regnault’s, 113 

Pitch of tones, 365; methods of determining, 365; standard, 370 

Planté, secondary cell of, 292 

Plateau, experiments of, in capillarity, 97 


INDEX. 521 


Plates, rise of liquid between, 100; transverse vibrations of, 376 

Poggendorff, explanation of gyroscope, 55 

Poisseuille, friction in liquids, 88 

Poisson, correction for use of piezometer, 113; theory of magnetism, 244 

Polariscope, 481, 482 

Polarization, of an electrolyte, 287; of cells, 291 

Polarization of light, by double refraction, 476; by reflection, 480; plane of, 
480; by refraction, 480; by reflection from fine particles, 481; elliptic and 
circular, 486; circular by reflection, 489; rotation of plane of, by quartz, 
491; by liquids, 493; in magnetic field, 493 

Polarized light, 478; explanation of, 478; effects of plates of doubly refracting 
crystals on, 483, 486, 488, 489, 491 

Polarizer, 481 

Polarizing angle, 480 

Pole, magnetic, 224, 228; unit magnetic, 226 

Poles, of a voltaic cell, 290 

Porous body, 103 

Potential, difference of, 34; absolute, 35, 37; within spherical shell, 38; outside 
sphere, 39. 

Potential, electrical, in a closed conductor, 249, 252; of a conductor, 252; zero, 
positive, and negative, 253; of asystem of conductors, 261; difference of, 
measured, 267 

Potential, magnetic, due to bar magnet, 228; due to magnetic shell, 232; of a 
closed circuit is multiply-valued, 306; illustrated by Faraday, 306 

Potentiometer, Clark’s, 334 

Pressure, 108, 109; in a fluid, 120; modified by outside forces, 121; surfaces of 
equal, 121; diminished on walls containing moving liquid column, 134 

Principal plane of crystal, 475 

Prism, 407 

Projectiles, path of, 81; movement of, in circle, 82 

Properties of matter, 4 

Pulley, 46 

Pump, 132; air, 137; compressing, 140 


QUALITY of tones, 365, 380; dependent upon harmonic tones, 380; upon change 
of phase, 381 
Quarter wave plates, 488 
Quartz, effects of plates of, in polarized light, 491; imitation of, 492 
Quincke, change in volume of dielectric, 263; electrical endosmose, 288; move- 
ments of electrolyte, 288; theory of electrolysis, 289 


RADIANT ENERGY, effects of, 462; transmission and absorption of, 466; emission 
of, 468; origin of, 469 


522 INDEX. 

Radiation, 167; intensity of, as dependent on distance, 463; on angle of in- 
cidence, 464; kind of, as dependent on temperature, 470 

Radicals, chemical, 85 

Radiometer, 192 

Rainbow, 457; secondary, 458 

Ratio between electrostatic and electromagnetic units, 336; a velocity, 336, 
physical significance of, 337, 338 

Ray of light, 422 

Rayleigh, electromotive force of Clark's cell, 293 

Reeds, in organ pipes, 373; lips used as, 374; vocal chords as, 374 

Reflection, of waves, 362; law of, 363; of light, law of, 404; total, 408; of 
spherical waves, 424; selective, 467; polarization of light by, 480 

Refraction of light, law of, 405; angle of, 406; dependent on wave length, 409; 
at spherical surfaces, 415; polarization of light by, 480; conical, 490 

Regelation, 177 

Regnault, his piezometer, 113; expansion of mercury, 170; extension of Du- 
long and Petit’s law, 176; modification of Dalton’s law, 182; modification 
of Gay-Lussac’s law, 185; pressure of water vapor, 186; total heat of 
steam, 201 

Resistance, electrical, 276, 329; depends on circuit, 276; of homogeneous cyl- 
inder, 278; specific, 278; varies with temperature, 279; units of, 329; boxes, 
331; measurement of, 332; of a divided circuit, 333; used to measure tem- 
perature, 149. 

Resonator, 382 

Reusch, artificial quartzes, 492 

Reuss, electrical endosmose, 288 

Rider, hot-air engine, 216 

Rigidity, 114; modulus of, 114 

Rods, longitudinal vibrations of, 374; transverse vibrations of, 376 

Roemer, determination of velocity of light, 432 

Rotation of plane of polarization by quartz, 491; right-handed and left-handed, 
492; by liquids, 493; in magnetic field, 493; explanation of, 494; by reflec- 
tion from magnet, 495 

Rotational coefficient, Hall’s, 316 

Rowland, mechanical equivalent of heat, 159, 205; magnetic permeability, 239; 
force on magnet due to moving electrical charge, 337; measurement of v, 
338; photographs of solar spectrum, 452; curved grating, 452 

Ruhmkorff’s coil, 328 

Rumford, relation of heat and energy, 145; views concerning work done by 
animals, 219 


SACCHARIMETER, 493 
Saturation of a magnet, 243 


INDEX. 523 


Savart, his toothed wheel, 365 

Scales, musical, 368; transposition of, 369; tempered, 370 

Schénbein, chemical theory of electromotive force, 293 

Schweigger, his multiplier, 316 

Screens, diffraction effects at, 445 

Screw, 48 

Seebeck, thermo-electric currents, 340; thermo-electric series, 341 

Self-induction, 325 

Set, 118 

Shadows, optical, 402 

Shear, 108, 113; amount of, 114; axis of, 114 

Shearing stress, 108; strain, 109 

Shunt circuit, 334 

Siphon, 131 

Siren, determination of number of vibrations by 366 

’ Smee’s cell, 291 

Snell, law of refraction, 406 

Solenoid, 314 

Solidification, 176 

Solids, 85; structure of, 86; crystalline, amorphous, 86; movements of, due to 
capillarity, 102; distinction between fluids and, 119; soft, hard, 119 

Solubility, 104 

Solution, 103 

Sound, 353; origin of, 353; propagation of, 354; theoretical velocity of, 390; 
velocity of, in air, 392; measurements, 394 

Sounding boards, 378 

Specific gravity, 125; determination of, for solids, 125; for liquids, 126; for 
gases, 127; correction for temperature, 174 

Specific gravity bottle, 126 

Specific heat, 152; mean, 153; varies with temperature, 175; with change of 
state, 175 

Specific heat of gases, 194; at constant volume, I94; at constant pressure, 
194; ratio of these, 195; determination of, at constant pressure, 195; rela- 
tion to elasticities, 198 

Specific inductive capacity. See Capacity, specific inductive, 257 

Spectrometer, 451; method of using, 451 

Spectroscope, 459 

Spectrum, pure, 447; produced by diffraction grating, 447; of first order, etce., 
447; formed by prism, 453; solar, 453, 458; dark lines in, 458; study of, 
459; of solids and liquids, 459; of gases, 460, explanation of, of a gas, 469; 
characteristics of, 471; of gases which undergo dissociation, 471 

Spectrum analysis, 459 


524 * INDEX. 


Spheroidal state, 183 

Spherometer, 8 . 

Spottiswoode and Moulton, electrical discharge in high vacua, 352 

Sprengel, his air-pump, 139; theory of, 134 

Statics, II 

Steam, total heat of, 201 

Stirling’s hot-air engine, 216 

Stokes, study of fluorescence, 472 

Strain, 108 

Stress, 29; in medium, 108 

Substances, simple, compound, 85 

Sun, energy of, 221 

Surface density of electrification, 251 

Surface energy of liquids, 93 

Surface tension of liquids, 91; relations to surface energy, 93; modified by 
electrical effects, 294 


TAIT, experiments in thermo-electricity, 342; thermo-electric formula, 345 

Telephonic transmitters and receivers, 327 

‘Telescope, 430, 431; magnifying power of, 431 

Temperament of musical scale, 370 

Temperature, 146; scales of, 147; change of, during solidification, 178; critical, 
183, 184; absolute zero of, 191; absolute, 212; movable equilibrium of, 
468; radiation of heat dependent on, 470 

Tension, 108; elasticity of, 114 

Thermodynamics, laws of, 205 

Thermo-electric currents, 340; how produced, 342; reversal of, 342 

Thermo-electric diagram, 342 

Thermo-electric element, 341 

‘Thermo-electric power, 343 

Thermo-electrically positive and negative, 341 

Thermometer, 146; construction of, 146; air, 149; limits in range of, 149; 
weight, 149, 172; registering, 150 

Thermopile, 341; used to measure temperature, 149 

Thomson, vortices, 135; absolute scale of temperature, 214; theory of solar 
energy, 221; treatment of magnetic induction, 238; magnetic permeability, 
239; absolute electrometer, 265; quadrant electrometer, 267; law of coun- 
ter electromotive force, 280; contact theory of electromotive force, 293; 
measurement of v, 336; thermo-electric currents in non-homogeneous cir- 
cuits, 342; thermo-electric power a function of temperature, 345; the 
Thomson effect, 347; electromotive force required to force spark through 


air, 349 


§ 


INDEX. 525 


Thomson effect, 345 

Tides, energy of, 220 

Time, unit of, 8; measurements of, 8, 9; Maxwell’s proposed unit of, 83 

Tones, musical, 365; differences in, 365; determination of number of vibra- 
tions in, 365; whole and semi-, 369; fundamental, 373; analysis of complex, 
382; resultant, 387 

Tonic, 369 . 

Torricelli, barometer, 122; experiment of, 123; theorem for velocity of efflux, 
131; experiments to prove, 133 

Torsion, amount of, 116; moment of, 116 

Torsion balance, 116, 249 

Transmission of radiations, 466 

Triad, major, 368; minor, 368 

Tubes, rise of liquid in capillary, 99; drops in capillary, 101 

Tuning-fork, 376; sounding-box of, 378 


UMBRA, 402 
Units, fundamental and derived, 4; dimensions of, 10; systems of, Io 


VACUUM TUBE, electrical discharge in, 350 

Vapor, 180; saturated, 180; pressure of, 181; production of, in limited space, 
183; departure of, from Boyle’s law, 185; pressure of saturated, 186; pres- 
sure of water-, 186; in air, determination of, 202; pressure of, 202 

Vaporization, energy necessary for, 200; heat equivalent of, 200 

Velocity, 13; angular, 48; constant in a circle, 61 

Velocity of efflux of a liquid, 129; into a vacuum, 133 

Velocity, mean, of molecules of gas, 193 

Velocities, composition and resolution of, 16; of angular, 49 

Vena contracta, 134 

Ventral segment, 362 

Verdet, electromagnetic rotation of plane of polarization, 493 

Vernier, 5 

Vertex of spherical mirror, 411 

Vibrations of sounding bodies, 371; modes of exciting in tubes, 373; longi- 
tudinal, of rods, 374; of cords, 374; transverse, of cords, 375; of rods, 376; 
of plates, 376; communication of, 377; of a membrane, 378; optical 
method of studying, 383; velocity of propagation of, 390 

Vibrations, light, transverse to ray, 478; relation to plane of polarization, 480; 
elliptical and circular, 486 

Viscosity, 88; of solids, 119 


526 INDEX. 


Vision, ancient theory of, 396; Aristotle’s view of, 396 

Visual angle, 429 

Vocal chords, 374 

Volt, a unit of electromotive force, 326 

Volta, change in volume of Leyden jar, 263; electrophorus, 269; contact differ- 
ence of potential, 272; voltaic battery, 273; heating by current, 273; con- 
tact theory of electromotive force, 293 

Voltaic cells, 290; polarization of, 291; theories of electromotive force of, 293; 
arrangements of, 335 

Voltaic cells, kinds of : Grove’s gas battery, 290; Smee’s, 291; Daniell’s, 291; 
Grove’s, 292; Planté’s secondary, 292; Clark’s, 293 

Voltameter, weight, 285; volume, 285 

Volume, change of, with change of state, 177 

Vortex, in perfect fluid, 135; line, 135; filament, 135; properties of a, 136; 
strength of, 136; illustrations of, 137 

Vowel sounds, dependent on quality, 383 


WATER, specific heat of, 151; maximum density of, 161, 174; expansion of, by 
heat, 174; on solidification, 177 

Water-power, energy of, 218 

Wave, simple, 23; compound, 24, 359; propagation of, 354; length, 355; pro- 
gressive, 355 

Wave, sound, 356; mode of propagation of, 356; graphic representation of, 356; 
displacement in, 358; velocity of vibration in, 359; stationary, 361; reflec- 
tion of, 362; in sounding bodies, 371 

Wave, light, surface of, 397; relation of, to the direction of propagation, 401;. 
emergent from prism, 407, 408; measurement of length of, 440, 450; values 
of lengths of, 452; surface of, in uniaxial crystals, 476; in biaxial crystals, 
490° 

Weber, theory of magnetism, 244; theory of diamagnetism, 315; his electro- 
dynamometer, 321 

Weber and Kohlrausch, measurement of 2, 336 

Wedge, 48 

Weighing, methods of, 78 

Weight of a body, 70 

Wheatstone, his bridge, 331 

Wheel and axle, 47 

Wiedemann, electrical endosmose, 288 

Wind power, energy of, 218 

Wollaston, dark lines in solar spectrum, 458 

Work, 31; and energy, equivalence of, 32; unit of, 32 


INDEX. 527 


Wren, theory of gravitation, 68 
Wright, connection of electromotive force and heat of chemical combination, 
286 


YOUNG, theory of capillarity, 91; modulus of elasticity, 115; optical method of 
studying vibrations, 383; interference of light from two similar sources, 
439 


ank. 


a: 


ht ee eo 
an. a we 


a 


ne 


ii 


NIVERSITY OF ILLINOIS-URBA 


U 


NA 


3 0112 067638855 


